Datasets:

roseking
/

openAI-math-dataset

import Mathlib

namespace OAI

/-!
Finite Fortuin–Kasteleyn maps and the Brownian continuum random tree.

All combinatorial maps below have a distinguished dart 0 and are quotiented
by orientation-preserving dart relabellings fixing that dart. A rotation system
with Euler characteristic 2 encodes precisely a connected oriented-sphere map.
No simplicity restriction is imposed. Vertices and edges are rotation and
edge-reversal orbits, respectively. Every edge subset is included in the law.
The index n here represents source size n+1, so the limit is through ALL
positive edge sizes, not a subsequence.

MetricMeasureData packages the actual distance, measurable space and measure,
without using arbitrary representatives of measure-preserving isometry classes
as a global typeclass. Valid objects are compact metric probability spaces.
The GHP extended distance is the infimum over common isometric embeddings,
represented by pseudometrics on the disjoint union; zero cross distances simply
identify points in the common ambient space. Tests are bounded functions
continuous for that distance on valid objects. Thus GHP convergence is written
in its bounded-continuous-test (weak distributional convergence) definition.

A standard normalized Brownian excursion is represented in its standard
three-dimensional Bessel-bridge realization: the norm of three independent
standard Brownian bridges of duration 1. No rescaling of the excursion is used.
-/

noncomputable section
open Set Filter MeasureTheory ProbabilityTheory
open scoped Topology BigOperators ENNReal NNReal

namespace FKCRT

abbrev Dart (n : ℕ) := Fin (2 * (n + 1))

abbrev Orbit {n : ℕ} (s : Equiv.Perm (Dart n)) :=
  Quotient (Equiv.Perm.SameCycle.setoid s)

def orbit {n : ℕ} (s : Equiv.Perm (Dart n)) (d : Dart n) : Orbit s :=
  Quotient.mk _ d

instance {n : ℕ} (s : Equiv.Perm (Dart n)) : Fintype (Orbit s) := Fintype.ofFinite _

/-- The spanning graph with precisely the edges in A, with loops omitted only
from adjacency: they still exist as edge orbits, have FK weight, and contribute
two darts to the degree measure. Omitting loops does not change distance or k. -/
def spanningGraph {n : ℕ} (a s : Equiv.Perm (Dart n)) (A : Set (Orbit a)) :
    SimpleGraph (Orbit s) :=
  SimpleGraph.fromRel fun v w =>
    ∃ d : Dart n, orbit a d ∈ A ∧ orbit s d = v ∧ orbit s (a d) = w

/-- Labelled connected orientable genus-zero rotation systems. -/
def MapConditions {n : ℕ} (P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n)) : Prop :=
  Function.Involutive P.1 ∧ (∀ d, P.1 d ≠ d) ∧
  (spanningGraph P.1 P.2 Set.univ).Connected ∧
  Nat.card (Orbit P.2) + Nat.card (Orbit (P.2 * P.1)) = n + 3

abbrev LabelledMap (n : ℕ) :=
  {P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n) // MapConditions P}

/-- Root-preserving isomorphism of oriented maps (both dart permutations intertwine). -/
def MapIso {n : ℕ} (M N : LabelledMap n) : Prop :=
  ∃ r : Equiv.Perm (Dart n), r 0 = 0 ∧
    (∀ d, N.val.1 (r d) = r (M.val.1 d)) ∧
    (∀ d, N.val.2 (r d) = r (M.val.2 d))

abbrev RootedMap (n : ℕ) := Quotient (Relation.EqvGen.setoid (@MapIso n))

instance (n : ℕ) : Fintype (RootedMap n) := Fintype.ofFinite _

def reversal {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.1

def rotation {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.2

abbrev Vertex {n : ℕ} (M : RootedMap n) := Orbit (rotation M)
abbrev Edge {n : ℕ} (M : RootedMap n) := Orbit (reversal M)

def graph {n : ℕ} (M : RootedMap n) (A : Set (Edge M)) : SimpleGraph (Vertex M) :=
  spanningGraph (reversal M) (rotation M) A

/-- The source k_M(A), including isolated vertices. -/
def clusterCount {n : ℕ} (M : RootedMap n) (A : Finset (Edge M)) : ℕ :=
  Nat.card (graph M (A : Set (Edge M))).ConnectedComponent

/-- Exact decorated finite-map weight from eq:finite-law. -/
def fkWeight {n : ℕ} (q : ℝ) (M : RootedMap n) (A : Finset (Edge M)) : ℝ :=
  Real.rpow q ((clusterCount M A : ℝ) +
    ((A.card : ℝ) - (Nat.card (Vertex M) : ℝ)) / 2)

def partition (q : ℝ) (n : ℕ) : ℝ :=
  ∑ M : RootedMap n, ∑ A : Finset (Edge M), fkWeight q M A

/-- Unbundled presentation of a measured metric space; validity is specified below. -/
structure MetricMeasureData where
  Carrier : Type
  distance : Carrier → Carrier → ℝ
  measurableSpace : MeasurableSpace Carrier
  mass : @Measure Carrier measurableSpace

/-- Exactly compact metric probability spaces, with their Borel sigma algebras. -/
def MetricMeasureData.Valid (X : MetricMeasureData) : Prop :=
  ∃ m : MetricSpace X.Carrier,
    letI := m
    (∀ x y, dist x y = X.distance x y) ∧
    CompactSpace X.Carrier ∧ X.measurableSpace = borel X.Carrier ∧
    X.mass Set.univ = 1

/-- Normalized degree measure is uniform dart mass pushed to vertices. -/
def degreeMass {n : ℕ} (M : RootedMap n) :
    @Measure (Vertex M) ⊤ :=
  letI : MeasurableSpace (Vertex M) := ⊤
  ∑ d : Dart n, ENNReal.ofReal (1 / (2 * (n + 1) : ℝ)) •
    Measure.dirac (orbit (rotation M) d)

/-- Root and decoration are not included in the metric probability space. -/
def mapSpace {n : ℕ} (scale : ℝ) (M : RootedMap n) : MetricMeasureData where
  Carrier := Vertex M
  distance v w := scale * ((graph M Set.univ).dist v w : ℝ)
  measurableSpace := ⊤
  mass := degreeMass M

/-- An ambient pseudometric on the disjoint union giving isometric embeddings. -/
def Amalgamates (X Y : MetricMeasureData)
    (m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : Prop :=
  letI := m
  (∀ x x', dist (Sum.inl x : X.Carrier ⊕ Y.Carrier) (Sum.inl x') = X.distance x x') ∧
  (∀ y y', dist (Sum.inr y : X.Carrier ⊕ Y.Carrier) (Sum.inr y') = Y.distance y y')

def ambientCost (X Y : MetricMeasureData)
    (m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : ℝ≥0∞ :=
  letI := m
  letI : MeasurableSpace (X.Carrier ⊕ Y.Carrier) := borel _
  max (Metric.hausdorffEDist (Set.range (Sum.inl : X.Carrier → X.Carrier ⊕ Y.Carrier))
    (Set.range (Sum.inr : Y.Carrier → X.Carrier ⊕ Y.Carrier)))
    (levyProkhorovEDist
      (@Measure.map X.Carrier (X.Carrier ⊕ Y.Carrier) X.measurableSpace
        (borel _) Sum.inl X.mass)
      (@Measure.map Y.Carrier (X.Carrier ⊕ Y.Carrier) Y.measurableSpace
        (borel _) Sum.inr Y.mass))

/-- The common-isometric-embedding definition of the source GHP distance. -/
def ghpEDist (X Y : MetricMeasureData) : ℝ≥0∞ :=
  ⨅ m : {m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier) // Amalgamates X Y m},
    ambientCost X Y m.val

/-- Bounded continuous tests on compact metric probability spaces modulo
measure-preserving isometry, expressed on presentations. -/
def IsGHPTest (F : MetricMeasureData → ℝ) : Prop :=
  (∃ C : ℝ, ∀ X, |F X| ≤ C) ∧
  ∀ X, X.Valid → ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
    ∀ Y, Y.Valid → ghpEDist X Y < ENNReal.ofReal δ → |F Y - F X| < ε

/-- Source excursion pseudometric, with the closed interval minimum. -/
def excursionDist (e : unitInterval → ℝ) (s t : unitInterval) : ℝ :=
  e s + e t - 2 * sInf (e '' Set.uIcc s t)

abbrev ExcursionTree (e : unitInterval → ℝ) :=
  Quotient (Relation.EqvGen.setoid (fun s t => excursionDist e s t = 0))

def treeProjection (e : unitInterval → ℝ) (t : unitInterval) : ExcursionTree e :=
  Quotient.mk _ t

/-- The quotient by d_e=0, its induced metric, and pushforward of uniform time.
The equivalence closure is redundant for continuous e; the induced distance
is independent of representatives, as required by the pseudometric proof. -/
def treeSpace (e : unitInterval → ℝ) : MetricMeasureData :=
  letI : TopologicalSpace (ExcursionTree e) :=
    TopologicalSpace.coinduced (treeProjection e) inferInstance
  letI : MeasurableSpace (ExcursionTree e) := borel _
  { Carrier := ExcursionTree e
    distance := fun x y => excursionDist e x.out y.out
    measurableSpace := borel _
    mass := Measure.map (treeProjection e) (volume : Measure unitInterval) }

/-- Standard normalized excursion, realized as the 3D Bessel bridge. -/
def standardExcursion {Ω : Type} (B : Fin 3 → ℝ≥0 → Ω → ℝ) (ω : Ω)
    (t : unitInterval) : ℝ :=
  Real.sqrt (∑ i : Fin 3, (B i ⟨(t : ℝ), t.property.1⟩ ω -
    (t : ℝ) * B i 1 ω) ^ 2)

/-- Exact expectation under the finite law, with n+1 edges. -/
def finiteExpectation (q scale : ℝ) (n : ℕ) (F : MetricMeasureData → ℝ) : ℝ :=
  ∑ M : RootedMap n, ∑ A : Finset (Edge M),
    (fkWeight q M A / partition q n) * F (mapSpace scale M)

def brownianExpectation {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω)
    (B : Fin 3 → ℝ≥0 → Ω → ℝ) (F : MetricMeasureData → ℝ) : ℝ :=
  ∫ ω, F (treeSpace (standardExcursion B ω)) ∂P

/-- thm:main. The quantifiers over Brownian realizations express that the limit
is the standard excursion-tree law, not a coupling-specific object. -/
def MainStatement : Prop :=
  ∀ q : ℝ, 4 < q → ∃ c : ℝ, 0 < c ∧
    ∀ (Ω : Type) (mΩ : MeasurableSpace Ω) (P : @Measure Ω mΩ),
      @IsProbabilityMeasure Ω mΩ P →
      ∀ B : Fin 3 → ℝ≥0 → Ω → ℝ,
        (∀ i, @IsBrownianReal Ω mΩ (B i) P) →
        @iIndepFun Ω (Fin 3) mΩ (fun _ => ℝ≥0 → ℝ) (fun _ => inferInstance)
          (fun i ω t => B i t ω) P →
        ∀ F : MetricMeasureData → ℝ, IsGHPTest F →
          Tendsto (fun n : ℕ => finiteExpectation q (c / Real.sqrt (n + 1)) n F)
            atTop (𝓝 (@brownianExpectation Ω mΩ P B F))

end FKCRT
end

end OAI

	import Mathlib

	namespace OAI

	/-!
	Finite Fortuin–Kasteleyn maps and the Brownian continuum random tree.

	All combinatorial maps below have a distinguished dart 0 and are quotiented
	by orientation-preserving dart relabellings fixing that dart. A rotation system
	with Euler characteristic 2 encodes precisely a connected oriented-sphere map.
	No simplicity restriction is imposed. Vertices and edges are rotation and
	edge-reversal orbits, respectively. Every edge subset is included in the law.
	The index n here represents source size n+1, so the limit is through ALL
	positive edge sizes, not a subsequence.

	MetricMeasureData packages the actual distance, measurable space and measure,
	without using arbitrary representatives of measure-preserving isometry classes
	as a global typeclass. Valid objects are compact metric probability spaces.
	The GHP extended distance is the infimum over common isometric embeddings,
	represented by pseudometrics on the disjoint union; zero cross distances simply
	identify points in the common ambient space. Tests are bounded functions
	continuous for that distance on valid objects. Thus GHP convergence is written
	in its bounded-continuous-test (weak distributional convergence) definition.

	A standard normalized Brownian excursion is represented in its standard
	three-dimensional Bessel-bridge realization: the norm of three independent
	standard Brownian bridges of duration 1. No rescaling of the excursion is used.
	-/

	noncomputable section
	open Set Filter MeasureTheory ProbabilityTheory
	open scoped Topology BigOperators ENNReal NNReal

	namespace FKCRT

	abbrev Dart (n : ℕ) := Fin (2 * (n + 1))

	abbrev Orbit {n : ℕ} (s : Equiv.Perm (Dart n)) :=
	Quotient (Equiv.Perm.SameCycle.setoid s)

	def orbit {n : ℕ} (s : Equiv.Perm (Dart n)) (d : Dart n) : Orbit s :=
	Quotient.mk _ d

	instance {n : ℕ} (s : Equiv.Perm (Dart n)) : Fintype (Orbit s) := Fintype.ofFinite _

	/-- The spanning graph with precisely the edges in A, with loops omitted only
	from adjacency: they still exist as edge orbits, have FK weight, and contribute
	two darts to the degree measure. Omitting loops does not change distance or k. -/
	def spanningGraph {n : ℕ} (a s : Equiv.Perm (Dart n)) (A : Set (Orbit a)) :
	SimpleGraph (Orbit s) :=
	SimpleGraph.fromRel fun v w =>
	∃ d : Dart n, orbit a d ∈ A ∧ orbit s d = v ∧ orbit s (a d) = w

	/-- Labelled connected orientable genus-zero rotation systems. -/
	def MapConditions {n : ℕ} (P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n)) : Prop :=
	Function.Involutive P.1 ∧ (∀ d, P.1 d ≠ d) ∧
	(spanningGraph P.1 P.2 Set.univ).Connected ∧
	Nat.card (Orbit P.2) + Nat.card (Orbit (P.2 * P.1)) = n + 3

	abbrev LabelledMap (n : ℕ) :=
	{P : Equiv.Perm (Dart n) × Equiv.Perm (Dart n) // MapConditions P}

	/-- Root-preserving isomorphism of oriented maps (both dart permutations intertwine). -/
	def MapIso {n : ℕ} (M N : LabelledMap n) : Prop :=
	∃ r : Equiv.Perm (Dart n), r 0 = 0 ∧
	(∀ d, N.val.1 (r d) = r (M.val.1 d)) ∧
	(∀ d, N.val.2 (r d) = r (M.val.2 d))

	abbrev RootedMap (n : ℕ) := Quotient (Relation.EqvGen.setoid (@MapIso n))

	instance (n : ℕ) : Fintype (RootedMap n) := Fintype.ofFinite _

	def reversal {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.1

	def rotation {n : ℕ} (M : RootedMap n) : Equiv.Perm (Dart n) := M.out.val.2

	abbrev Vertex {n : ℕ} (M : RootedMap n) := Orbit (rotation M)
	abbrev Edge {n : ℕ} (M : RootedMap n) := Orbit (reversal M)

	def graph {n : ℕ} (M : RootedMap n) (A : Set (Edge M)) : SimpleGraph (Vertex M) :=
	spanningGraph (reversal M) (rotation M) A

	/-- The source k_M(A), including isolated vertices. -/
	def clusterCount {n : ℕ} (M : RootedMap n) (A : Finset (Edge M)) : ℕ :=
	Nat.card (graph M (A : Set (Edge M))).ConnectedComponent

	/-- Exact decorated finite-map weight from eq:finite-law. -/
	def fkWeight {n : ℕ} (q : ℝ) (M : RootedMap n) (A : Finset (Edge M)) : ℝ :=
	Real.rpow q ((clusterCount M A : ℝ) +
	((A.card : ℝ) - (Nat.card (Vertex M) : ℝ)) / 2)

	def partition (q : ℝ) (n : ℕ) : ℝ :=
	∑ M : RootedMap n, ∑ A : Finset (Edge M), fkWeight q M A

	/-- Unbundled presentation of a measured metric space; validity is specified below. -/
	structure MetricMeasureData where
	Carrier : Type
	distance : Carrier → Carrier → ℝ
	measurableSpace : MeasurableSpace Carrier
	mass : @Measure Carrier measurableSpace

	/-- Exactly compact metric probability spaces, with their Borel sigma algebras. -/
	def MetricMeasureData.Valid (X : MetricMeasureData) : Prop :=
	∃ m : MetricSpace X.Carrier,
	letI := m
	(∀ x y, dist x y = X.distance x y) ∧
	CompactSpace X.Carrier ∧ X.measurableSpace = borel X.Carrier ∧
	X.mass Set.univ = 1

	/-- Normalized degree measure is uniform dart mass pushed to vertices. -/
	def degreeMass {n : ℕ} (M : RootedMap n) :
	@Measure (Vertex M) ⊤ :=
	letI : MeasurableSpace (Vertex M) := ⊤
	∑ d : Dart n, ENNReal.ofReal (1 / (2 * (n + 1) : ℝ)) •
	Measure.dirac (orbit (rotation M) d)

	/-- Root and decoration are not included in the metric probability space. -/
	def mapSpace {n : ℕ} (scale : ℝ) (M : RootedMap n) : MetricMeasureData where
	Carrier := Vertex M
	distance v w := scale * ((graph M Set.univ).dist v w : ℝ)
	measurableSpace := ⊤
	mass := degreeMass M

	/-- An ambient pseudometric on the disjoint union giving isometric embeddings. -/
	def Amalgamates (X Y : MetricMeasureData)
	(m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : Prop :=
	letI := m
	(∀ x x', dist (Sum.inl x : X.Carrier ⊕ Y.Carrier) (Sum.inl x') = X.distance x x') ∧
	(∀ y y', dist (Sum.inr y : X.Carrier ⊕ Y.Carrier) (Sum.inr y') = Y.distance y y')

	def ambientCost (X Y : MetricMeasureData)
	(m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier)) : ℝ≥0∞ :=
	letI := m
	letI : MeasurableSpace (X.Carrier ⊕ Y.Carrier) := borel _
	max (Metric.hausdorffEDist (Set.range (Sum.inl : X.Carrier → X.Carrier ⊕ Y.Carrier))
	(Set.range (Sum.inr : Y.Carrier → X.Carrier ⊕ Y.Carrier)))
	(levyProkhorovEDist
	(@Measure.map X.Carrier (X.Carrier ⊕ Y.Carrier) X.measurableSpace
	(borel _) Sum.inl X.mass)
	(@Measure.map Y.Carrier (X.Carrier ⊕ Y.Carrier) Y.measurableSpace
	(borel _) Sum.inr Y.mass))

	/-- The common-isometric-embedding definition of the source GHP distance. -/
	def ghpEDist (X Y : MetricMeasureData) : ℝ≥0∞ :=
	⨅ m : {m : PseudoMetricSpace (X.Carrier ⊕ Y.Carrier) // Amalgamates X Y m},
	ambientCost X Y m.val

	/-- Bounded continuous tests on compact metric probability spaces modulo
	measure-preserving isometry, expressed on presentations. -/
	def IsGHPTest (F : MetricMeasureData → ℝ) : Prop :=
	(∃ C : ℝ, ∀ X, \|F X\| ≤ C) ∧
	∀ X, X.Valid → ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
	∀ Y, Y.Valid → ghpEDist X Y < ENNReal.ofReal δ → \|F Y - F X\| < ε

	/-- Source excursion pseudometric, with the closed interval minimum. -/
	def excursionDist (e : unitInterval → ℝ) (s t : unitInterval) : ℝ :=
	e s + e t - 2 * sInf (e '' Set.uIcc s t)

	abbrev ExcursionTree (e : unitInterval → ℝ) :=
	Quotient (Relation.EqvGen.setoid (fun s t => excursionDist e s t = 0))

	def treeProjection (e : unitInterval → ℝ) (t : unitInterval) : ExcursionTree e :=
	Quotient.mk _ t

	/-- The quotient by d_e=0, its induced metric, and pushforward of uniform time.
	The equivalence closure is redundant for continuous e; the induced distance
	is independent of representatives, as required by the pseudometric proof. -/
	def treeSpace (e : unitInterval → ℝ) : MetricMeasureData :=
	letI : TopologicalSpace (ExcursionTree e) :=
	TopologicalSpace.coinduced (treeProjection e) inferInstance
	letI : MeasurableSpace (ExcursionTree e) := borel _
	{ Carrier := ExcursionTree e
	distance := fun x y => excursionDist e x.out y.out
	measurableSpace := borel _
	mass := Measure.map (treeProjection e) (volume : Measure unitInterval) }

	/-- Standard normalized excursion, realized as the 3D Bessel bridge. -/
	def standardExcursion {Ω : Type} (B : Fin 3 → ℝ≥0 → Ω → ℝ) (ω : Ω)
	(t : unitInterval) : ℝ :=
	Real.sqrt (∑ i : Fin 3, (B i ⟨(t : ℝ), t.property.1⟩ ω -
	(t : ℝ) * B i 1 ω) ^ 2)

	/-- Exact expectation under the finite law, with n+1 edges. -/
	def finiteExpectation (q scale : ℝ) (n : ℕ) (F : MetricMeasureData → ℝ) : ℝ :=
	∑ M : RootedMap n, ∑ A : Finset (Edge M),
	(fkWeight q M A / partition q n) * F (mapSpace scale M)

	def brownianExpectation {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω)
	(B : Fin 3 → ℝ≥0 → Ω → ℝ) (F : MetricMeasureData → ℝ) : ℝ :=
	∫ ω, F (treeSpace (standardExcursion B ω)) ∂P

	/-- thm:main. The quantifiers over Brownian realizations express that the limit
	is the standard excursion-tree law, not a coupling-specific object. -/
	def MainStatement : Prop :=
	∀ q : ℝ, 4 < q → ∃ c : ℝ, 0 < c ∧
	∀ (Ω : Type) (mΩ : MeasurableSpace Ω) (P : @Measure Ω mΩ),
	@IsProbabilityMeasure Ω mΩ P →
	∀ B : Fin 3 → ℝ≥0 → Ω → ℝ,
	(∀ i, @IsBrownianReal Ω mΩ (B i) P) →
	@iIndepFun Ω (Fin 3) mΩ (fun _ => ℝ≥0 → ℝ) (fun _ => inferInstance)
	(fun i ω t => B i t ω) P →
	∀ F : MetricMeasureData → ℝ, IsGHPTest F →
	Tendsto (fun n : ℕ => finiteExpectation q (c / Real.sqrt (n + 1)) n F)
	atTop (𝓝 (@brownianExpectation Ω mΩ P B F))

	end FKCRT
	end

	end OAI