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f61e547 | 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 | import OAI.Probability.FreeIsing.Trees.InwardCutChannels
import OAI.Probability.FreeIsing.Trees.InwardCutFields
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.InwardCutChannels
open PathSideBranches CutContinuationChannels ConnectedHull
def inwardContinuationProduct {d k L : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0) (theta : ℝ)
(p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) (s : Finset ℕ)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) : ℝ :=
∏ i ∈ s, responseAttenuation theta
(cutInwardGaussianField μ t (appendCutContinuation p w).1 (k + 1) i z) ^ 2
def inwardContinuationAverage {d k : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0) (theta : ℝ)
(p : TreeShell d (k + 1)) (L : ℕ) (s : Finset ℕ)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) : ℝ :=
(∑ w : ReducedTail (cutLastColor p) L, inwardContinuationProduct μ t theta p w s z) /
(Fintype.card (ReducedTail (cutLastColor p) L) : ℝ)
theorem inwardContinuationProduct_Ico_eq_range {d k L : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0) (theta : ℝ)
(p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) :
inwardContinuationProduct μ t theta p w (Finset.Ico (k + 1) (k + 1 + L)) z =
∏ j ∈ Finset.range L, responseAttenuation theta
(cutInwardGaussianField μ t (appendCutContinuation p w).1
(k + 1) (k + 1 + j) z) ^ 2 := by
simp only [inwardContinuationProduct, Finset.prod_Ico_eq_prod_range,
Nat.add_sub_cancel_left]
theorem takeVertex_appendCutContinuation {d k L : ℕ}
(p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) :
takeVertex (appendCutContinuation p w).1 (k + 1) = p.1 := by
apply Subtype.ext
simp [takeVertex, appendCutContinuation, p.2]
private theorem inward_region_subset_cut {d k L : ℕ}
(p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) (i : ℕ) :
cone (takeVertex (appendCutContinuation p w).1 (k + 1)) \
cone (takeVertex (appendCutContinuation p w).1 (i + 1)) ⊆ cone p.1 := by
rw [takeVertex_appendCutContinuation]
exact Set.sdiff_subset
private theorem reducedTail_nonempty {d k : ℕ} (hd : 2 ≤ d)
(p : TreeShell d (k + 1)) (L : ℕ) :
Nonempty (ReducedTail (cutLastColor p) L) := by
apply Fintype.card_pos_iff.mp
rw [card_reducedTail]
exact pow_pos (by omega : 0 < d - 1) L
theorem exists_inwardContinuationAverage_channel {d k : ℕ} (hd : 2 ≤ d)
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0)
{theta : ℝ} (hθ : theta ^ 2 ≤ 1)
(p : TreeShell d (k + 1)) (L : ℕ) (s : Finset ℕ) :
∃ g : ConeSample ℝ p.1 → ℝ, Measurable g ∧
(∀ z, g z ∈ Set.Icc (0 : ℝ) 1) ∧
∀ z, inwardContinuationAverage μ t theta p L s z = g (globalCone p.1 z) := by
let := reducedTail_nonempty hd p L
have hF : Measurable (Function.uncurry (fun a x => (t : ℝ) * spinSign a + x)) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul
(t : ℝ)).add measurable_snd
exact exists_regionAverage_channel
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) hF hθ p.1
(fun w i => cone (takeVertex (appendCutContinuation p w).1 (k + 1)) \
cone (takeVertex (appendCutContinuation p w).1 (i + 1)))
(fun w i => takeVertex (appendCutContinuation p w).1 i)
(fun _ => s) (inward_region_subset_cut p)
theorem integral_inwardContinuationAverage_le {d k : ℕ} (hd : 2 ≤ d)
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0)
{theta : ℝ} (hθ : theta ^ 2 ≤ 1)
(p : TreeShell d (k + 1)) (L : ℕ) (s : Finset ℕ)
(P : Measure ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)))
[IsProbabilityMeasure P] {m : ℝ}
(hmean : ∀ w : ReducedTail (cutLastColor p) L,
(∫ z, inwardContinuationProduct μ t theta p w s z ∂P) ≤ m) :
(∫ z, inwardContinuationAverage μ t theta p L s z ∂P) ≤ m := by
let := reducedTail_nonempty hd p L
have hF : Measurable (Function.uncurry (fun a x => (t : ℝ) * spinSign a + x)) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul
(t : ℝ)).add measurable_snd
exact integral_regionAverage_le
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) hF hθ p.1
(fun w i => cone (takeVertex (appendCutContinuation p w).1 (k + 1)) \
cone (takeVertex (appendCutContinuation p w).1 (i + 1)))
(fun w i => takeVertex (appendCutContinuation p w).1 i)
(fun _ => s) (inward_region_subset_cut p) P hmean
end Problem367.InwardCutChannels
end
end OAI
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