File size: 4,070 Bytes
c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Fields.FieldMagnetization
import OAI.Probability.IsingPerceptron.CascadePair
namespace OAI
/-! The common-prefix depth of the actual finite cascade has the field
partition's interval lengths. The statement allows arbitrary tied values. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
namespace InvariantIsing
lemma fieldLevelIndex_tail_iff (h : FieldStep) (i : Fin h.depth) (s : ℝ) :
i.val + 1 ≤ (fieldLevelIndex h s).val ↔ h.cut i.succ.castSucc ≤ s := by
constructor
· intro hi
by_contra hs
have hb : fieldLevelIndex h s ≤ i.castSucc := by
apply Finset.sup_le
intro j _
split_ifs with hj
· by_contra hji
have hij : i.succ.castSucc ≤ j.castSucc := by
change i.val + 1 ≤ j.val
have hh : i.val < j.val := lt_of_not_ge hji
omega
exact hs ((h.ordered_cut.monotone hij).trans hj)
· exact bot_le
change (fieldLevelIndex h s).val ≤ i.val at hb
omega
· intro hs
have hi := Finset.le_sup (s := Finset.univ)
(f := fun j : Fin (h.depth + 1) => if h.cut j.castSucc ≤ s then j else 0)
(Finset.mem_univ i.succ)
have hh : i.succ ≤ fieldLevelIndex h s := by
simpa only [hs, ite_true, fieldLevelIndex] using hi
exact hh
def fieldCommonLevel (h : FieldStep) (σ : ℕ → LabeledLeaf h.depth) : Fin (h.depth + 1) :=
⟨labeledCommonDepth h.depth (σ 0) (σ 1),
Nat.lt_succ_of_le (labeledCommonDepth_le h.depth (σ 0) (σ 1))⟩
theorem fieldCommonLevel_integral (h : FieldStep) (a : Fin (h.depth + 1) → ℝ) :
(∫ σ, a (fieldCommonLevel h σ)
∂cascadeReplicaLaw h.depth (chainExponent h.cut)) =
∫ s, a (fieldLevelIndex h s) ∂pathMeasure := by
let L := fun s => (fieldLevelIndex h s).val
have hLm : Measurable L :=
(measurable_of_countable (Fin.val : Fin (h.depth + 1) → ℕ)).comp
(fieldLevelIndex_monotone h).measurable
have hLn : ∀ᵐ s ∂pathMeasure, L s ≤ h.depth :=
ae_of_all _ fun s => Nat.le_of_lt_succ (fieldLevelIndex h s).isLt
have hT (i : Fin h.depth) : pathMeasure.real {s | i.val + 1 ≤ L s} =
1 - chainExponent h.cut i := by
have he : {s | i.val + 1 ≤ L s} = Ici (h.cut i.succ.castSucc) := by
ext s
exact fieldLevelIndex_tail_iff h i s
rw [he]
have hc : h.cut i.succ.castSucc ∈ Icc (0 : ℝ) 1 := by
constructor
· rw [← h.first]
exact h.ordered_cut.monotone (Fin.zero_le _)
· rw [← h.last]
exact h.ordered_cut.monotone (Fin.le_last _)
have hb : chainExponent h.cut i = h.cut i.succ.castSucc := by
rw [chainExponent_apply h.cut i.isLt]
rfl
rw [hb]
exact unitUniform_Ici_real hc
let a' := fun k : ℕ => a ⟨min k h.depth, Nat.lt_succ_of_le (min_le_right _ _)⟩
have hlaw := cascade_pair_law_of_tails pathMeasure h.depth (chainExponent h.cut)
(chainExponent_admissible h.ordered_cut h.first h.last) a' hLm hLn hT
have hmleft : Measurable (fun s => a' (L s)) := (measurable_of_countable a').comp hLm
have hmright : Measurable (fun σ : ℕ → LabeledLeaf h.depth =>
cascadeScalarArray h.depth a' σ 0 1) := by
exact (measurable_pi_apply 1).comp
((measurable_pi_apply 0).comp (measurable_cascadeScalarArray h.depth a'))
have he := congrArg (fun μ : Measure ℝ => ∫ x, x ∂μ) hlaw
rw [integral_map hmleft.aemeasurable (by fun_prop),
integral_map hmright.aemeasurable (by fun_prop)] at he
have hleft (s : ℝ) : a' (L s) = a (fieldLevelIndex h s) := by
dsimp only [a', L]
congr 1
apply Fin.ext
exact min_eq_left (Nat.le_of_lt_succ (fieldLevelIndex h s).isLt)
have hright (σ : ℕ → LabeledLeaf h.depth) :
cascadeScalarArray h.depth a' σ 0 1 = a (fieldCommonLevel h σ) := by
dsimp only [cascadeScalarArray, a', fieldCommonLevel]
congr 1
apply Fin.ext
exact min_eq_left (labeledCommonDepth_le h.depth (σ 0) (σ 1))
simpa only [hleft, hright] using he.symm
end InvariantIsing
end
end OAI
|