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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 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Angular.TwoRpowNMul
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFixedSpatialTerm
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
namespace Attempt4FullFar
open MeasureTheory Set Metric Filter Real GraphSourceScale
open scoped ContDiff RealInnerProductSpace
lemma graph_source_fixed_geometry (N a p v sig : ℝ) (G c₀ : ℝ)
(_hc₀ : 0 < c₀) :
let R := (2:ℝ)^N
let H := (2:ℝ)^(p-a)
let V := (2:ℝ)^(v-p)
let δ := min c₀ (H⁻¹*R^(10*sig))
let τ := (2:ℝ)^(-p)*R^(30*sig)
let D := G*(2:ℝ)^(-a)
((2:ℝ)^v*τ=V*R^(30*sig)) ∧
(D/τ=G*H*R^(-30*sig)) ∧
(δ=min c₀ ((2:ℝ)^a/(2:ℝ)^p*R^(10*sig))) := by
dsimp only
have ht : 0 < (2:ℝ)^N := by positivity
constructor
· rw [←mul_assoc,←Real.rpow_add (by norm_num)]; congr 2
constructor
· rw [div_eq_mul_inv,mul_inv_rev,←Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),neg_neg,
←Real.rpow_neg ht.le]
rw [show G*(2:ℝ)^(-a)*(((2:ℝ)^N)^(-(30*sig))*(2:ℝ)^p) =
G*((2:ℝ)^p*(2:ℝ)^(-a))*((2:ℝ)^N)^(-30*sig) by ring_nf]
rw [←Real.rpow_add (by norm_num)]; congr 2
· rw [←Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),neg_sub,
Real.rpow_sub (by norm_num)]
theorem graph_source_fixed_nonedge_decay
(V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
{sig c₀ Ccap G : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀)
(hCcap : 0 ≤ Ccap) (hG : 0 ≤ G) (T : ℝ) :
∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (a p v L : ℝ), p-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p →
let R := (2:ℝ)^N
let δ := min c₀ (((2:ℝ)^(p-a))⁻¹*R^(10*sig))
let τ := (2:ℝ)^(-p)*R^(30*sig)
let A := 1+R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))+δ⁻¹
∀ (Ψ : ℝ → ℂ) (w₁ w₂ d₁ d₂ e₁ e₂ : V) (K B₁ B₂ : ℝ) (b₁ b₂ : V → ℂ),
ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 →
(∫ u : ℝ, ‖Ψ u‖) ≤ K → (∀ u, ‖iteratedDeriv m Ψ u‖ ≤ K) →
‖w₁‖=1 → ‖w₂‖=1 → 0 ≤ K → 0 ≤ B₁ → 0 ≤ B₂ →
‖e₁‖ ≤ G*(2:ℝ)^(-a) → ‖e₂‖ ≤ G*(2:ℝ)^(-a) →
‖e₁-d₁‖ ≤ G*(2:ℝ)^(-p) → ‖e₂-d₂‖ ≤ G*(2:ℝ)^(-p) →
¬(‖d₁-inner ℝ d₁ w₁ • w₁‖ ≤ τ ∧
‖d₂-inner ℝ d₂ w₂ • w₂‖ ≤ τ ∧ |inner ℝ w₁ d₁+inner ℝ w₂ d₂| ≤ τ) →
ContDiff ℝ ∞ b₁ → ContDiff ℝ ∞ b₂ →
(∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j b₁ w‖ ≤ B₁*A^j) →
(∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j b₂ w‖ ≤ B₂*A^j) →
(∀ w, ‖w‖=1 → b₁ w ≠ 0 → ‖w-w₁‖ ≤ Ccap*δ) →
(∀ w, ‖w‖=1 → b₂ w ≠ 0 → ‖w-w₂‖ ≤ Ccap*δ) →
‖graphFixedSpatialTermND Ψ ((2:ℝ)^v) e₁ e₂ b₁ b₂‖ ≤ C*K*B₁*B₂*R^(-T) := by
obtain ⟨m,hm⟩ := exists_nat_gt (T/(29*sig))
have hm' : T ≤ 29*sig*m := by
have hh := (div_lt_iff₀ (by positivity : 0 < 29*sig)).mp hm
linarith
obtain ⟨B,hB,hBb⟩ := graph_fixed_nonedge_term_uniform V m
have hbase : 0 ≤ 5+c₀⁻¹+G := by positivity
refine ⟨m,B*(5+c₀⁻¹+G)^m,by positivity,?_⟩
have hpow : Tendsto (fun N : ℕ => (2:ℝ)^N) atTop atTop :=
tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
obtain ⟨N₀,hN₀⟩ := Filter.eventually_atTop.mp
(((tendsto_rpow_atTop (by positivity : 0 < 20*sig)).comp hpow).eventually
(Filter.eventually_ge_atTop (8*Ccap*G+4*G)))
refine ⟨N₀,?_⟩
intro N hN a p v L hh hL hq R δ τ A Ψ w₁ w₂ d₁ d₂ e₁ e₂ K B₁ B₂ b₁ b₂
hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hB₁ hB₂ he₁ he₂ hed₁ hed₂ hne hb₁ hb₂ hd₁ hd₂ hs₁ hs₂
have hR : 1 ≤ R := one_le_pow₀ (by norm_num)
have hRp : 0 < R := lt_of_lt_of_le zero_lt_one hR
have hδ : 0 < δ := lt_min hc₀ (by positivity)
have hτ : 0 < τ := by dsimp [τ]; positivity
have hA : 1 ≤ A := by
have : 0 ≤ R^sig*max 1 ((2:ℝ)^L*R^(-50*sig)) := by positivity
dsimp [A]; linarith [inv_pos.mpr hδ]
have hD : 0 ≤ G*(2:ℝ)^(-a) := by positivity
have hη : 0 ≤ G*(2:ℝ)^(-p) := by positivity
have hg := graph_source_fixed_geometry (N:ℝ) a p v sig G c₀ hc₀
simp only [Real.rpow_natCast] at hg
rcases hg with ⟨hQτ,hDτ,hδeq⟩
have herr : 4*(Ccap*δ)*(G*(2:ℝ)^(-a))+2*(G*(2:ℝ)^(-p)) ≤ τ/2 := by
dsimp only [δ,τ,R]
rw [hδeq]
simp only [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),←div_eq_mul_inv]
exact source_error_scale hR hsig.le (by positivity) (by positivity) hCcap hG (hN₀ N hN)
have hterm := hBb Ψ w₁ w₂ d₁ d₂ e₁ e₂ ((2:ℝ)^v) τ (Ccap*δ)
(G*(2:ℝ)^(-a)) (G*(2:ℝ)^(-p)) A K B₁ B₂ b₁ b₂
hΨ hsΨ hK hdΨ hw₁ hw₂ (by positivity) hτ (by positivity) hD hη hA
hK0 hB₁ hB₂ he₁ he₂ hed₁ hed₂ herr hne hb₁ hb₂ hd₁ hd₂ hs₁ hs₂
obtain ⟨hH,hU⟩ := length_scale hh hL
simp only [Real.rpow_natCast] at hH hU
have hV : 1 ≤ (2:ℝ)^(v-p) := Real.one_le_rpow (by norm_num) hq
have hratio := ratio_power_decay hc₀ hR hsig.le (by positivity : 0 < (2:ℝ)^(p-a)) hV
hG (by positivity : 0 ≤ (2:ℝ)^L) hH hU hm'
have heq : (A+1+G*(2:ℝ)^(-a)/τ)/((2:ℝ)^v*τ) =
(2+R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))+δ⁻¹+
G*(2:ℝ)^(p-a)*R^(-30*sig))/((2:ℝ)^(v-p)*R^(30*sig)) := by
rw [hQτ,hDτ]; dsimp [A]; ring
rw [heq] at hterm
apply hterm.trans
calc
_ ≤ B*K*B₁*B₂*((5+c₀⁻¹+G)^m*R^(-T)) := mul_le_mul_of_nonneg_left hratio (by positivity)
_ = _ := by ring
end Attempt4FullFar
end
end
end OAI
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