File size: 4,420 Bytes
f205175 | 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 | import OAI.Probability.DilutedSpin.ZeroShape
namespace OAI
section
section
namespace DilutedSpinGlass.UniversalDictionary
open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory HeterogeneousMarks PrescribedTree ConcreteReservoir Filter Set
open scoped NNReal BigOperators Topology
variable {L p : ℕ}
lemma physicalShape_zero (m : Fin (L+1) → ℝ)
(hm : ∀ j, 0 < m j) (hmono : Monotone m) (hend : m (Fin.last L) = 1)
(S : PrescribedTree (L+1)) (a : S.Leaf)
(M : Model p) (C H : ℝ) (N : ℕ) (u : Spec L×ℕ → ℝ)
(F : (Option (Fin 0) → Option (Fin 0) → ℕ) → ℝ) (spinAnchor : Bool)
(f : (S.Leaf → FinitePath (Fin N → Spin) (L+1)) → ℝ)
{B : ℝ} (hB : 0 ≤ B) (hf : ∀ x, |f x| ≤ B) :
physicalShapeCovariance m S a M C H N u F spinAnchor f =
F (fun _ _ => L+1) * physicalCoefficient (weights L) prior m direction anchor
S a M C H N u f (zeroSpec L spinAnchor) 0 := by
exact shapeCovariance_zero (fun _ : Fin N => M.field.toMeasure) (bondLaw M N)
(markLaw (weights L) N) (siteLaw N) (M.alpha*N) (scoreRate N) S a
(KernelTower.terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L)
(Fin.cons 0 m) (physicalBase M C H N)
(dictionaryFactor (observableAt direction N) (observableAt anchor N) u)
(fun v x => readSpin (KernelTower.terminalState L x) v) F spinAnchor f
(measurable_physicalBase M C H N) (fun j => (hm j).ne')
(exponent_cons_monotone m (fun j => (hm j).le) hmono)
(exponent_cons_nonneg m (fun j => (hm j).le)) (by simp) hend hB hf
theorem complete_shape_selection (m : Fin (L+1) → ℝ)
(M : Model p) {C H c : ℝ} (hC : 0 ≤ C) (hH : 0 ≤ H) (hc : 0 < c)
(hθ : ∀ᵐ z ∂M.disorder.toMeasure, ∀ σ, |z.1 σ| ≤ C)
(hh : ∀ᵐ h ∂M.field.toMeasure, |h| ≤ H)
(hθi : Integrable (fun z : InteractionSample p => ‖z.1‖) M.disorder.toMeasure)
(hhi : Integrable (fun h : ℝ => |h|) M.field.toMeasure)
(hm : ∀ l, c ≤ m l) (hmono : Monotone m) (hend : m (Fin.last L) = 1)
{ε : ℝ} (hε : 0 < ε) :
∃ (Ns : ℕ → ℕ) (us : ℕ → Spec L×ℕ → ℝ), StrictMono Ns ∧
(∀ n i, us n i ∈ Icc (probeLow i.2) (probeHigh i.2)) ∧
(∀ n, ConcreteReservoir.increment (weights L) prior m direction anchor M (Ns n) (us n) ≤
liminf (pressure M) atTop+ε) ∧
(∀ (S : PrescribedTree (L+1)) (a : S.Leaf)
(f : (n : ℕ) → (S.Leaf → FinitePath (Fin (Ns n+1) → Spin) (L+1)) → ℝ)
(B : ℝ), 0 ≤ B → (∀ n x, |f n x| ≤ B) →
∀ (k : ℕ) (F : (Option (Fin k) → Option (Fin k) → ℕ) → ℝ) (spinAnchor : Bool),
Tendsto (fun n => physicalShapeCovariance m S a M C H (Ns n+1) (us n)
F spinAnchor (f n)) atTop (𝓝 0)) := by
obtain ⟨Ns,us,hNs,hus,hinc,he,hcoef⟩ := universal_analytic_selection m M hC hH hc
hθ hh hθi hhi hm hmono hend hε
refine ⟨Ns,us,hNs,hus,hinc,?_⟩
intro S a f B hB hf k F spinAnchor
by_cases hk : k = 0
· subst k
simp_rw [physicalShape_zero m (fun j => hc.trans_le (hm j)) hmono hend S a M C H _ _
F spinAnchor _ hB (hf _)]
simpa using (hcoef S a f B hB hf (zeroSpec L spinAnchor) 0).const_mul (F (fun _ _ => L+1))
have hk' : 0 < k := Nat.pos_of_ne_zero hk
simp_rw [physicalShape_expansion hk' m (fun j => hc.trans_le (hm j)) hmono hend
S a M C H _ _ F spinAnchor _ hB (hf _)]
have hh (r : ℕ) (b c : Fin r → Option (Fin k)) (d : Fin r → Fin L) :
Tendsto (fun n => physicalHistoryCovariance m S a M C H (Ns n+1) (us n)
b c d spinAnchor (f n)) atTop (𝓝 0) := by
simp_rw [physicalHistory_polarization hk' m (fun j => hc.trans_le (hm j)) hmono hend
S a M C H _ _ b c d spinAnchor _ hB (hf _)]
have ht := tendsto_finsetSum (Finset.univ : Finset (Fin k → Bool))
(fun e _ => (hcoef S a f B hB hf (polarizedSpec hk' e b c d spinAnchor) k).const_mul
(∏ j, Polarization.sign (e j)))
simpa using ht.const_mul ((k:ℝ)^k / 2^k)
have ht := tendsto_finsetSum (Finset.univ : Finset (Finset (SharingIndex L (Option (Fin k)))))
(fun s _ => (hh _ (fun j => (requirementMap s j).1) (fun j => (requirementMap s j).2.1)
(fun j => (requirementMap s j).2.2)).const_mul
(BooleanPolynomial.coefficient (fun y => F (splitFromProfile y)) s))
simpa using ht
end DilutedSpinGlass.UniversalDictionary
end
end
end OAI
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