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f1f7953 | 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 | import OAI.Analysis.IrrationalRotation.Sampling.BlockPhaseConstruction
import OAI.Analysis.IrrationalRotation.Product.ProductBlockCost
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped BigOperators Topology
namespace Invariant233.ProductModel
open ProductPhases
theorem exists_admissible_phase_block_weight_and
(m : ℕ → ℕ) (hgrowth : ∀ j, 2 ^ j ≤ m j)
(K G C : ℕ → ℝ) (hK : ∀ j, 1 ≤ K j) (hG : ∀ j, 0 < G j)
(hC : ∀ j, (K j) ^ 2 ≤ C j)
(hscale : ∀ j, 2 < (m j : ℝ) * K j)
(hsum : Summable (fun j => 1 / K j))
(e : ℕ ≃ Σ j, Fin (m j))
(he : ∀ a : Σ j, Fin (m j),
e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val)
{P : (ℕ → ℝ) → Prop}
(hP : ∀ᵐ θ ∂Measure.infinitePi (fun i => ProbabilityTheory.cond volume
(Icc (1 / ((m (e i).1 : ℝ) * K (e i).1))
(2 / ((m (e i).1 : ℝ) * K (e i).1)))), P θ) :
∃ θ : ℕ → ℝ,
(∀ j, ∀ i ∈ consecutiveBlock m j,
θ i ∈ Icc (1 / ((m j : ℝ) * K j)) (2 / ((m j : ℝ) * K j))) ∧
(∑' i, θ i) ≤ 2 * (∑' j, 1 / K j) ∧
AdmissibleAnglesAndCosts θ (fun i => C (e i).1) ∧
HasNegInfiniteLogIntegral baseMeasure (weight θ (fun i => C (e i).1)) ∧
(∀ᵐ x ∂baseMeasure,
0 < weight θ (fun i => C (e i).1) x ∧ weight θ (fun i => C (e i).1) x ≤ 1) ∧
(∀ j, smallPhaseCondition (consecutiveBlock m j) (K j) (G j) θ) ∧
(∀ᶠ j in atTop,
largePhaseCondition (consecutiveBlock m j) ((m j : ℝ) * K j) θ) ∧ P θ := by
have hm : ∀ j, 0 < m j := fun j =>
lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j)
have hKpos (j : ℕ) : 0 < K j := zero_lt_one.trans_le (hK j)
have hCpos (j : ℕ) : 0 < C j := (sq_pos_of_pos (hKpos j)).trans_le (hC j)
obtain ⟨θ, hθ, hpos, hsθ, htθ, hind, hsmall, hlarge, hp⟩ :=
exists_consecutive_phase_angles_and m hgrowth K G hKpos hG hsum e he hP
have hbounds (i : ℕ) : θ i ∈
Icc (1 / ((m (e i).1 : ℝ) * K (e i).1))
(2 / ((m (e i).1 : ℝ) * K (e i).1)) :=
hθ (e i).1 i (mem_consecutiveBlock_enumeration m e he i)
have hunit (i : ℕ) : θ i < 1 :=
(hbounds i).2.trans_lt ((div_lt_one
(mul_pos (Nat.cast_pos.mpr (hm _)) (hKpos _))).mpr (hscale _))
refine ⟨θ, hθ, htθ, ?_, ?_, ?_, hsmall, hlarge, hp⟩
· exact ⟨fun i => ⟨hpos i, hunit i, hCpos _⟩, hsθ, hind⟩
· exact hasNegInfiniteLogIntegral_of_block_bounds m hm K C hK hC e θ
(fun i => ⟨(hpos i).le, hunit i⟩) (fun i => (hbounds i).1)
· filter_upwards [ae_weight_pos θ (fun i => C (e i).1)
(fun i => (hpos i).le) (fun i => (hCpos _).le) hsθ] with x hx
exact ⟨hx, (weight_mem_Icc θ _ x).2⟩
theorem exists_admissible_phase_block_weight
(m : ℕ → ℕ) (hgrowth : ∀ j, 2 ^ j ≤ m j)
(K G C : ℕ → ℝ) (hK : ∀ j, 1 ≤ K j) (hG : ∀ j, 0 < G j)
(hC : ∀ j, (K j) ^ 2 ≤ C j)
(hscale : ∀ j, 2 < (m j : ℝ) * K j)
(hsum : Summable (fun j => 1 / K j)) :
∃ (e : ℕ ≃ Σ j, Fin (m j)) (θ : ℕ → ℝ),
(∀ a : Σ j, Fin (m j),
e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val) ∧
(∀ j, ∀ i ∈ consecutiveBlock m j,
θ i ∈ Icc (1 / ((m j : ℝ) * K j)) (2 / ((m j : ℝ) * K j))) ∧
(∑' i, θ i) ≤ 2 * (∑' j, 1 / K j) ∧
AdmissibleAnglesAndCosts θ (fun i => C (e i).1) ∧
HasNegInfiniteLogIntegral baseMeasure (weight θ (fun i => C (e i).1)) ∧
(∀ᵐ x ∂baseMeasure,
0 < weight θ (fun i => C (e i).1) x ∧ weight θ (fun i => C (e i).1) x ≤ 1) ∧
(∀ j, smallPhaseCondition (consecutiveBlock m j) (K j) (G j) θ) ∧
(∀ᶠ j in atTop,
largePhaseCondition (consecutiveBlock m j) ((m j : ℝ) * K j) θ) := by
have hm : ∀ j, 0 < m j := fun j =>
lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j)
obtain ⟨e, he⟩ := InvariantProjection.IndependentAngles.exists_consecutiveBlocks_equiv m hm
obtain ⟨θ, hθ, htθ, hadm, hlog, hweight, hsmall, hlarge, _⟩ :=
exists_admissible_phase_block_weight_and m hgrowth K G C hK hG hC hscale hsum e he
(P := fun _ => True) (Filter.Eventually.of_forall fun _ => True.intro)
exact ⟨e, θ, he, hθ, htθ, hadm, hlog, hweight, hsmall, hlarge⟩
end Invariant233.ProductModel
end
end OAI
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