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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.PathSideBranches
theorem prefixFlip_mem_branch_iff {d : ℕ} (c : Fin d) (u : TreeVertex d) :
[c] <+: (prefixFlip c u).1 ↔ ¬ [c] <+: u.1 := by
rcases u with ⟨w, hw⟩
cases w with
| nil => simp [prefixFlip, prefixFlipWord]
| cons a w =>
by_cases h : c = a
· subst a
cases w with
| nil => simp [prefixFlip, prefixFlipWord]
| cons b w =>
have hcb : c ≠ b := (List.isChain_cons_cons.mp hw).1
simp [prefixFlip, prefixFlipWord, hcb]
· simp [prefixFlip, prefixFlipWord, h]
theorem exists_reroot_cone_complement {d : ℕ} (r : TreeVertex d) (hr : r.1 ≠ []) :
∃ (g : TreeAut d) (c : Fin d),
g.toEquiv r = ConnectedHull.root d ∧
g.toEquiv (parentVertex r) = ⟨[c], List.IsChain.singleton c⟩ ∧
∀ u : TreeVertex d, u ∉ cone r ↔ [c] <+: (g.toEquiv u).1 := by
obtain ⟨g, c, hp, hc, hgeo⟩ := exists_reroot_cone r hr
refine ⟨g.trans (prefixTreeAut c), c, ?_, ?_, ?_⟩
· change prefixFlip c (g.toEquiv r) = _
rw [hc]
apply Subtype.ext
simp [prefixFlip, prefixFlipWord, ConnectedHull.root]
· change prefixFlip c (g.toEquiv (parentVertex r)) = _
rw [hp]
rfl
· intro u
change u ∉ cone r ↔ [c] <+: (prefixFlip c (g.toEquiv u)).1
rw [prefixFlip_mem_branch_iff, hgeo]
end Problem367.PathSideBranches
namespace Problem367
open PathSideBranches
theorem selectedPosterior_image_relabel_ae
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [MeasurableSpace B]
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(ν : Measure A) [IsProbabilityMeasure ν]
{F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F))
(g : TreeAut d) (S : Set (TreeVertex d)) (r : TreeVertex d) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
selectedPosterior P F (Subtype.val : g.toEquiv '' S → TreeVertex d) (g.toEquiv r) ∘
Prod.map (relabel g) (relabel g) =ᵐ[P]
selectedPosterior P F (Subtype.val : S → TreeVertex d) r := by
have heq : Set.range (fun i : g.toEquiv '' S => g.toEquiv.symm i.1) =
Set.range (Subtype.val : S → TreeVertex d) := by
ext u
constructor
· rintro ⟨⟨v, ⟨x, hx, rfl⟩⟩, rfl⟩
exact ⟨⟨x, hx⟩, by simp⟩
· rintro ⟨x, rfl⟩
exact ⟨⟨g.toEquiv x.1, ⟨x.1, x.2, rfl⟩⟩, by simp⟩
have hfield := selectedObservationField_eq_of_range_eq F heq
have h := selectedPosterior_relabel_ae hμ ν hF
(Subtype.val : g.toEquiv '' S → TreeVertex d) (g.toEquiv r) g
simpa only [selectedPosterior, hfield, Equiv.symm_apply_apply] using h
theorem HasFreeIsingLaw.selectedPosterior_child_subset_complement
{d : ℕ} {theta : ℝ} {A B : Type*}
[MeasurableSpace A] [StandardBorelSpace A]
[MeasurableSpace B] [StandardBorelSpace B] [Nonempty B]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1)
(ν : Measure A) [IsProbabilityMeasure ν]
{F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F))
(r : TreeVertex d) (hr : r.1 ≠ [])
(T : Set (TreeVertex d)) (hT : T ⊆ (cone r)ᶜ) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
selectedPosterior P F (Subtype.val : T → TreeVertex d) r =ᵐ[P]
fun z => theta * selectedPosterior P F
(Subtype.val : T → TreeVertex d) (parentVertex r) z := by
obtain ⟨g, c, hr0, hpc, hgeo⟩ := exists_reroot_cone_complement r hr
let q : TreeVertex d := ⟨[c], List.IsChain.singleton c⟩
have hq : q.1 ≠ [] := by simp [q]
have hpq : parentVertex q = ConnectedHull.root d := by rfl
have hsub : g.toEquiv '' T ⊆ cone q := by
rintro u ⟨v, hv, rfl⟩
exact (hgeo v).mp (hT hv)
have hbase := hμ.selectedPosterior_parent_subset_cone h0 h1 ν hF q hq
(g.toEquiv '' T) hsub
have hmp := PlantedStationarity.measurePreserving_planted_relabel hμ ν g
have hb := hmp.quasiMeasurePreserving.ae_eq_comp hbase
have hleft := selectedPosterior_image_relabel_ae hμ ν hF g T r
have hright := selectedPosterior_image_relabel_ae hμ ν hF g T (parentVertex r)
rw [hr0] at hleft
rw [hpc] at hright
filter_upwards [hb, hleft, hright] with z hz hl hrz
rw [hpq] at hz
exact hl.symm.trans (hz.trans (congrArg (fun x : ℝ => theta * x) hrz))
theorem HasFreeIsingLaw.selected_gaussian_subset_complement_field_difference
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (r : TreeVertex d) (hr : r.1 ≠ [])
(S T U : Set (TreeVertex d)) (hS : S ⊆ cone r)
(hT : T ⊆ (cone r)ᶜ) (hU : U ⊆ (cone r)ᶜ)
(hp : r ∈ S) (ht : parentVertex r ∈ T) (hu : parentVertex r ∈ U) :
let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
let field := fun (V : Set (TreeVertex d)) (u : TreeVertex d) z =>
Real.artanh (selectedPosterior P F (Subtype.val : V → TreeVertex d) u z)
∀ᵐ z ∂P,
|field (S ∪ T) r z - field (S ∪ U) r z| ≤
Real.tanh beta * |field T (parentVertex r) z - field U (parentVertex r) z| ∧
|field (S ∪ T) r z - field (S ∪ U) r z| ≤ 2 * beta := by
obtain ⟨g, c, hr0, hpc, hgeo⟩ := exists_reroot_cone_complement r hr
let q : TreeVertex d := ⟨[c], List.IsChain.singleton c⟩
have hq : q.1 ≠ [] := by simp [q]
have hpq : parentVertex q = ConnectedHull.root d := by rfl
have hS' : g.toEquiv '' S ⊆ (cone q)ᶜ := by
rintro u ⟨v, hv, rfl⟩ hbad
exact ((hgeo v).mpr hbad) (hS hv)
have hT' : g.toEquiv '' T ⊆ cone q := by
rintro u ⟨v, hv, rfl⟩
exact (hgeo v).mp (hT hv)
have hU' : g.toEquiv '' U ⊆ cone q := by
rintro u ⟨v, hv, rfl⟩
exact (hgeo v).mp (hU hv)
have hp' : parentVertex q ∈ g.toEquiv '' S := ⟨r, hp, hr0.trans hpq.symm⟩
have ht' : q ∈ g.toEquiv '' T := ⟨parentVertex r, ht, hpc⟩
have hu' : q ∈ g.toEquiv '' U := ⟨parentVertex r, hu, hpc⟩
have hbase := hμ.selected_gaussian_subset_cone_field_difference hbeta t q hq
(g.toEquiv '' S) (g.toEquiv '' T) (g.toEquiv '' U) hS' hT' hU' hp' ht' hu'
have hmp := PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g
have hb := hmp.quasiMeasurePreserving.ae hbase
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
have hF : Measurable (Function.uncurry F) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul
(t : ℝ)).add measurable_snd
have hST := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g (S ∪ T) r
have hSU := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g (S ∪ U) r
have hTT := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g T (parentVertex r)
have hUU := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g U (parentVertex r)
filter_upwards [hb, hST, hSU, hTT, hUU] with z hz hst hsu htt huu
have hpc' : g.toEquiv (parentVertex r) = q := hpc
rw [hpq, ← hr0, ← hpc', ← Set.image_union, ← Set.image_union] at hz
simp only [Function.comp_apply] at hst hsu htt huu
dsimp only [F, PlantedSymmetry.gaussianNoiseLaw] at hst hsu htt huu hz ⊢
simpa only [hst, hsu, htt, huu] using hz
end Problem367
end
end OAI
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