| 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 |
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