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import OAI.Geometry.CylinderCovering.Physical.Finite
import OAI.Geometry.CylinderCovering.Radial.FinitePair
import OAI.Geometry.CylinderCovering.Physical.CostVariants
import OAI.Geometry.CylinderCovering.Physical.SweepTransport
namespace OAI
open Set Filter
open scoped Topology InnerProductSpace
noncomputable section
namespace RadialPhysical
open CylinderCovering
theorem physical_family_of_sweep_cover (centered : Bool) (b eta : ℝ)
(hb : 0 < b) (T : ℝ → ℝ) (hT : ContinuousOn T (Icc (-b) b))
(hT0 : ∀ s ∈ Icc (-b) b, 0 ≤ T s)
(hcover : modelBody centered b ⊆
upperSweepOn centered b eta T (Icc (-b) b) ∪
lowerSweepOn centered b eta T (Icc (-b) b))
(tagU tagL : (N : ℕ) → Fin N → ℝ)
(htagU : ∀ N, RadialRiemann.IsTagged (-b) b N (tagU N))
(htagL : ∀ N, RadialRiemann.IsTagged (-b) b N (tagL N)) :
∃ C : (N : ℕ) → Fin (2*N) → TriangularCovering.TriCylinder,
PhysicalFamilyConclusion centered b eta T tagU tagL C := by
obtain ⟨U, L, hUL, hlim, hbound, hsup⟩ :=
physical_finite centered b eta hb T hT hT0 tagU tagL htagU htagL
refine ⟨fun N => RadialPairedSweep.join (U N) (L N), ?_⟩
refine {
local_properties := ?_
covers_body := ?_
covers_sweeps := ?_
actual_covers_body := ?_
triangle_limit := RadialPairedSweep.join_totalArea_tendsto U L hlim
actual_bound := hbound
actual_limsup := hsup }
· intro N hN i
simpa only [RadialPairedSweep.join_upper, RadialPairedSweep.join_lower]
using (hUL N hN).1 i
· intro N hN z hz
exact RadialPairedSweep.join_covers_union (U N) (L N)
(fun z hz => ((hUL N hN).2.1 z hz).1)
(fun z hz => ((hUL N hN).2.2 z hz).1) z (hcover hz)
· intro N hN
exact RadialPairedSweep.join_covers_union (U N) (L N)
(fun z hz => ((hUL N hN).2.1 z hz).1)
(fun z hz => ((hUL N hN).2.2 z hz).1)
· intro N hN z hz
rcases hcover hz with hU | hL
· exact Or.inl (((hUL N hN).2.1 z hU).2)
· exact Or.inr (((hUL N hN).2.2 z hL).2)
theorem PhysicalFamilyConclusion.eventually_strict
{centered : Bool} {b eta : ℝ} {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)
{target : ℝ} (hsave : scaledCost b eta T < target) :
∀ᶠ N in atTop, 0 < N ∧ TriangularCovering.totalArea (C N) < target ∧
physicalActualCost centered b eta T N (tagU N) (tagL N) < target := by
have hgap : 0 < (target-scaledCost b eta T)/2 := by linarith
filter_upwards [RadialPairedSweep.eventually_strict_finite_cost C hC.triangle_limit hsave,
hC.actual_bound ((target-scaledCost b eta T)/2) hgap] with N hN hactual
exact ⟨hN.1, hN.2, by linarith⟩
lemma capChoice_continuous (square : Bool) (b eta beta : ℝ) :
Continuous (capChoice square b eta beta) := by
cases square <;> unfold capChoice <;> simp only [Bool.false_eq_true, ite_false, ite_true]
· unfold linearCap capPolynomial
fun_prop
· unfold squareCap capPolynomial
fun_prop
lemma capChoice_nonneg (square : Bool) (b eta beta s : ℝ) (hbeta : 0 ≤ beta) :
0 ≤ capChoice square b eta beta s := by
cases square
· simp only [capChoice, Bool.false_eq_true, ite_false]
unfold linearCap capPolynomial
positivity
· simp only [capChoice, ite_true]
exact Real.sqrt_nonneg _
lemma physicalModelCap_continuous (centered square : Bool) (b eta beta : ℝ) :
Continuous (physicalModelCap centered square b eta beta) := by
cases centered
· exact capChoice_continuous square b eta beta
· exact continuous_const.mul (capChoice_continuous square b eta beta)
lemma physicalModelCap_nonneg (centered square : Bool) (b eta beta s : ℝ)
(hbeta : 0 ≤ beta) : 0 ≤ physicalModelCap centered square b eta beta s := by
cases centered
· exact capChoice_nonneg square b eta beta s hbeta
· exact mul_nonneg (by norm_num) (capChoice_nonneg square b eta beta s hbeta)
lemma centered_linearCap_eq_double (b eta beta : ℝ) :
centeredLinearCap b eta beta = fun s => 2*linearCap b eta beta s := by
funext s
unfold centeredLinearCap linearCap capPolynomial
ring
theorem eventually_physicalModelCap_covers (centered square : Bool)
{b beta : ℝ} (hb : 0 < b) (hbeta : 0 < beta) :
∀ᶠ eta : ℝ in 𝓝[>] 0,
modelBody centered b ⊆
upperSweepOn centered b eta (physicalModelCap centered square b eta beta) (Icc (-b) b) ∪
lowerSweepOn centered b eta (physicalModelCap centered square b eta beta) (Icc (-b) b) := by
cases centered <;> cases square
· simpa only [physicalModelCap, capChoice, Bool.false_eq_true, ite_false] using
eventually_halfheight_linear_covers hb hbeta
· simpa only [physicalModelCap, capChoice, Bool.false_eq_true, ite_false, ite_true] using
eventually_halfheight_square_covers hb hbeta
· simpa only [physicalModelCap, capChoice, Bool.false_eq_true, ite_false, ite_true,
centered_linearCap_eq_double] using eventually_centered_linear_covers hb hbeta
· simpa only [physicalModelCap, capChoice, ite_true] using
eventually_centered_square_covers hb hbeta
private lemma scaledCost_double (b eta : ℝ) (T : ℝ → ℝ) :
scaledCost b eta (fun s => 2*T s) = 4*scaledCost b eta T := by
unfold scaledCost
have hd : (fun s => scaledDensity b eta (fun s => 2*T s) s) =
fun s => 4*scaledDensity b eta T s := by
funext s
unfold scaledDensity
ring
rw [hd, intervalIntegral.integral_const_mul]
ring
theorem eventually_physicalModelCap_strict (centered square : Bool)
{b beta : ℝ} (hb : 0 < b) (hbeta : 0 < beta) (hbetamax : beta < 1/60) :
∀ᶠ eta : ℝ in 𝓝[>] 0,
scaledCost b eta (physicalModelCap centered square b eta beta) <
physicalHalfReference centered b := by
have hhalf : ∀ᶠ eta : ℝ in 𝓝[>] 0,
scaledCost b eta (capChoice square b eta beta) < b*height b/2 := by
cases square
· exact scaled_linear_strict hb hbeta hbetamax
· exact scaled_square_strict hb hbeta hbetamax
cases centered
· exact hhalf
· filter_upwards [hhalf] with eta heta
change scaledCost b eta (fun s => 2*capChoice square b eta beta s) < 2*b*height b
rw [scaledCost_double]
linarith
theorem physical_cap_coverage : PhysicalCapCoverageStatement := by
intro centered square b beta hb hbeta
filter_upwards [eventually_physicalModelCap_covers centered square hb hbeta] with eta hcover
intro tagU tagL htagU htagL
exact 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
theorem physical_cap_finite : PhysicalCapFiniteStatement := 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_strict hcost⟩
end RadialPhysical
end
end OAI
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