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7f37411 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.Ledger
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Estimates.AssemblyInnerEnergyMem
namespace OAI
/-! Graph energies and pair estimates. -/
open _root_.OAI.PlanarFalconer
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport.CurrentFalconer.TreeND
open Set MeasureTheory Filter PlanarFalconer
open scoped ContDiff Classical
variable {n : ℕ}
lemma uniform_assembly_root_graph (Ctest : ℝ) {sig : ℝ} (hsig : 0<sig)
(hsig1 : sig≤1/100) {J : ℕ} (hJ : 0<J)
(W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))))
(B total : ℕ) (hW : ∀ L∈W,L.length≤B)
(Ψ : ℝ → ℝ) (hΨ : ContDiff ℝ ∞ Ψ) (hsupp : tsupport Ψ⊆Ioo (1/4:ℝ) 4)
(hΨ0 : ∀ r,0≤Ψ r) (ψ : EuclideanSpace ℝ (Fin (n+1)) → ℂ)
(hψ : ContDiff ℝ ∞ ψ) (hψ1 : ∀ w,‖w‖=1 → ‖ψ w‖≤1) :
∃ M : ℕ,∀ Cq : ℝ,0≤Cq → ∃ Cerr : ℝ,0≤Cerr ∧ ∃ N₀ : ℕ,
∀ e₁ e₂ : RecursiveEnvironmentND n,N₀≤e₁.N → e₂.N=e₁.N →
e₁.sig=sig → e₂.sig=sig → e₁.C=Ctest → e₂.C=Ctest →
e₁.J=J → e₂.J=J → 0<e₁.q → 0<e₂.q →
∀ B₀ s₁ s₂,RegularizedProfileData e₁.ρ e₁.N B₀ s₁ sig e₁.mass →
RegularizedProfileData e₂.ρ e₂.N B₀ s₂ sig e₂.mass →
e₁.ρ (treeCell [])=1 → e₂.ρ (treeCell [])=1 →
∀ c : ℕ,0<c → (c:ℝ)<(e₁.N:ℝ)/4 →
∀ (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))
(f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q P : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))),
Measurable f → Measurable g → (∀ x,‖f x‖≤1) → (∀ x,‖g x‖≤1) →
(∀ b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c,L₁ b∈W) →
(∀ b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c,L₂ b∈W) →
(∑ b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c,(L₁ b).length)≤total →
(∑ b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c,(L₂ b).length)≤total →
(∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) →
(∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j P.1 w‖≤Cq) →
assemblyRootEnergy e₁ e₂ c L₁ L₂ f g Q P Ψ ψ ≤
((2:ℝ)^e₁.N)^(-e₁.profile c-e₂.profile c+
min (height e₁.profile 0 c) (height e₂.profile 0 c)+(Ctest+36*(n+1:ℝ)+5)*sig)*
_root_.OAI.CurrentFalconer.TreeND.graphSourceChildMax e₁.ρ e₂.ρ 0 c e₁.N [] []
(((2:ℝ)^e₁.N)^(-e₁.mass c)) (((2:ℝ)^e₂.N)^(-e₂.mass c)) Ψ
(assemblySymbol e₁ c L₁ f Q) (assemblySymbol e₂ c L₂ g P)+
Cerr*((2:ℝ)^e₁.N)^(-2:ℝ) := by
let c₀ : ℝ := 1/(4*(n+2:ℝ))
have hc₀ : 0<c₀ := by dsimp [c₀]; positivity
have hcsmall : ((n+1:ℕ)+1:ℝ)*c₀≤1/2 := by
have he : ((n+1:ℕ)+1:ℝ)*c₀=1/4 := by
dsimp [c₀]; push_cast; field_simp; ring
rw [he]; norm_num
obtain ⟨M,C,hC,N₁,hgraph⟩ := _root_.OAI.CurrentFalconer.TreeND.graph_HF_finite_jet_estimate
(d:=n+1) hsig hc₀ hcsmall (le_refl (0:ℝ)) 2
obtain ⟨K,hK0,hK,hdK⟩ := _root_.OAI.CurrentFalconer.graph_fixed_radial_budget hΨ
(isCompact_Icc.of_isClosed_subset (isClosed_tsupport Ψ) (hsupp.trans Ioo_subset_Icc_self)) M
obtain ⟨Pcut,hPcut,hcut⟩ := _root_.OAI.CurrentFalconer.graph_fixed_angular_budget hψ M
refine ⟨M,?_⟩
intro Cq hCq
obtain ⟨E,hE,hjets⟩ := uniform_assemblySymbol_jets W M B total hW hsig Cq hCq
obtain ⟨N₂,hjets⟩ := Filter.eventually_atTop.mp hjets
obtain ⟨N₃,himposes⟩ := uniform_assemblySymbol_imposes (n:=n) hsig hJ
refine ⟨C*K*Pcut^2*E^4,by positivity,max N₁ (max N₂ N₃),?_⟩
intro e₁ e₂ hN heN hs₁ hs₂ hC₁ hC₂ hJ₁ hJ₂ hq₁ hq₂ B₀ s₁ s₂ hd₁ hd₂ hroot₁ hroot₂
c hcpos hc L₁ L₂ f g Q P hf hg hfn hgn hL₁ hL₂ ht₁ ht₂ hQ hP
let := hd₁.probability
let := hd₂.probability
have hNc : (c:ℝ)≤e₁.N := by have := Nat.cast_nonneg (α:=ℝ) e₁.N; linarith only [hc,this]
have hcN : c≤e₁.N := by exact_mod_cast hNc
have hcN₂ : c≤e₂.N := by omega
have h2c : 2*c≤e₁.N := by exact_mod_cast (show (2:ℝ)*c≤e₁.N by linarith only [hc,Nat.cast_nonneg (α:=ℝ) e₁.N])
have h2c₂ : 2*c≤e₂.N := by omega
have hlist₁ (b) (hb : b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c) : descriptorLength b≤e₁.N/2 := by
have hh := assemblyDescriptorList_length e₁.profile hq₁ h2c
have hb' := Finset.le_sup (s:=assemblyDescriptorList e₁.profile e₁.N e₁.q c) (f:=descriptorLength) hb
omega
have hlist₂ (b) (hb : b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c) : descriptorLength b≤e₁.N/2 := by
have hh := assemblyDescriptorList_length e₂.profile hq₂ h2c₂
have hb' := Finset.le_sup (s:=assemblyDescriptorList e₂.profile e₂.N e₂.q c) (f:=descriptorLength) hb
omega
have hj₁ := hjets e₁.N (by omega) e₁ rfl hs₁ hq₁ c (e₁.N/2) hcN hlist₁ L₁ f Q hf hfn hL₁ ht₁ hQ
have hj₂ := hjets e₁.N (by omega) e₂ heN hs₂ hq₂ c (e₁.N/2) (by omega) hlist₂ L₂ g P hg hgn hL₂ ht₂ hP
have hi₁ := himposes e₁ (by omega) hs₁ hJ₁ inferInstance hq₁ c hcN L₁ f Q
have hi₂ := himposes e₂ (by omega) hs₂ hJ₂ inferInstance hq₂ c hcN₂ L₂ g P
have hd₁' := (_root_.OAI.CurrentFalconer.TreeND.literal_regularizedProfileData_iff _ _ _ _ _ _).mp hd₁
have hd₂' := (_root_.OAI.CurrentFalconer.TreeND.literal_regularizedProfileData_iff _ _ _ _ _ _).mp hd₂
rw [heN] at hd₂'
have hroot₁' : 0<e₁.ρ (_root_.OAI.CurrentFalconer.TreeND.treeCell []) := by
have hr : 0<e₁.ρ (treeCell []) := by rw [hroot₁]; exact zero_lt_one
simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hr
have hroot₂' : 0<e₂.ρ (_root_.OAI.CurrentFalconer.TreeND.treeCell []) := by
have hr : 0<e₂.ρ (treeCell []) := by rw [hroot₂]; exact zero_lt_one
simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hr
have hrange : (c:ℝ)≤(e₁.N:ℝ)-c+2*sig*e₁.N := by
nlinarith [Nat.cast_nonneg (α:=ℝ) e₁.N]
have hL : ((e₁.N/2:ℕ):ℝ)≤(e₁.N:ℝ)-c+2*sig*e₁.N := by
have hh : ((e₁.N/2:ℕ):ℝ)≤(e₁.N:ℝ)/2 := by simpa using (Nat.cast_div_le (α:=ℝ) (m:=e₁.N) (n:=2))
nlinarith [Nat.cast_nonneg (α:=ℝ) e₁.N]
have hgap : 10*sig*e₁.N≤(e₁.N:ℝ)-c := by
nlinarith [Nat.cast_nonneg (α:=ℝ) e₁.N]
have hi₁' := hi₁
have hi₂' := hi₂
simp only [hC₁] at hi₁'
simp only [RecursiveEnvironmentND.profile,heN,hC₂] at hi₂'
have hh := hgraph e₁.N (by omega) e₁.ρ e₂.ρ B₀ s₁ s₂ e₁.mass e₂.mass hd₁' hd₂'
0 c e₁.N (e₁.N/2:ℕ) Ctest K E Pcut J 1 1 0 0 [] [] Ψ ψ
(assemblySymbol e₁ c L₁ f Q) (assemblySymbol e₂ c L₂ g P)
hcpos hNc le_rfl (by simpa using hrange) hL hgap (Nat.zero_le _) (Nat.zero_le _) hcpos hcpos
(by norm_num only [Nat.cast_zero,zero_add]; positivity) (by norm_num only [Nat.cast_zero,zero_add]; positivity) hJ hJ hJ
(by simp [_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths])
(by simp [_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths])
hroot₁' hroot₂' (by simp) hΨ (hsupp.trans Ioo_subset_Icc_self) hΨ0 hK (hdK M le_rfl) hK0
(by simpa only [Real.rpow_natCast] using hj₁)
(by simpa only [Real.rpow_natCast] using hj₂)
hPcut hψ hψ1 hcut hi₁' hi₂'
have hz₁ := assembly_profile_zero e₁ hd₁
have hz₂ := assembly_profile_zero e₂ hd₂
change _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₁.mass e₁.N 0=0 at hz₁
change _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₂.mass e₂.N 0=0 at hz₂
rw [heN] at hz₂
simp only [hz₁,hz₂,sub_zero,Nat.cast_zero,Nat.cast_add,Nat.cast_one] at hh
have hpow : -(e₁.profile c+e₂.profile c)+min (height e₁.profile 0 c) (height e₂.profile 0 c)+
(Ctest+36*(n+1:ℝ)+5)*sig = -e₁.profile c-e₂.profile c+min (height e₁.profile 0 c) (height e₂.profile 0 c)+
(Ctest+36*(n+1:ℝ)+5)*sig := by ring
have hp₁ : _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₁.mass e₁.N=e₁.profile := rfl
have hp₂ : _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₂.mass e₁.N=e₂.profile := by
rw [←heN]; rfl
simp only [hp₁,hp₂,hpow] at hh
simpa only [assemblyRootEnergy,heN,_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hh
end GraphLiteralSupport.CurrentFalconer.TreeND
end
end
end OAI
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