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namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
def wordPairCylinder {d : ℕ} (p : Word d × Word d) : Set (Path d × Path d) :=
wordCylinder p.1 ×ˢ wordCylinder p.2
lemma measurableSet_wordPairCylinder {d : ℕ} (p : Word d × Word d) :
MeasurableSet (wordPairCylinder p) :=
(measurableSet_pathCylinder _ _).prod (measurableSet_pathCylinder _ _)
lemma wordPair_transfer {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(x z : Site d) (a b : Word d) (h : Disjoint (wordDepartures a) (wordDepartures b)) :
annealed μ x (wordCylinder a) * annealed μ z (wordCylinder b) =
∫⁻ ω, quenchedKernel d (ω,x) (wordCylinder a) * quenchedKernel d (ω,z) (wordCylinder b)
∂environmentLaw μ := by
simp only [wordCylinder,annealed_cylinder,quenchedKernel_cylinder]
split_ifs <;> try simp only [zero_mul,mul_zero,lintegral_zero]
exact (path_weights_disjoint μ a.1 b.1 (wordPath a) (wordPath b) (by
intro i hi j hj hij
exact Set.disjoint_left.mp h ⟨i,hi,rfl⟩ ⟨j,hj,hij.symm⟩)).symm
noncomputable def freshPair {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(x z : Site d) : Measure (Environment d × (Path d × Path d)) :=
environmentLaw μ ⊗ₘ (freshPathKernel x ×ₖ freshPathKernel z)
instance {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x z : Site d) :
IsProbabilityMeasure (freshPair μ x z) := by
unfold freshPair
infer_instance
lemma freshPair_apply {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x z : Site d)
{E : Set (Environment d × (Path d × Path d))} (hE : MeasurableSet E) :
freshPair μ x z E = ∫⁻ ω, ((quenchedKernel d (ω,x)).prod (quenchedKernel d (ω,z)))
(Prod.mk ω ⁻¹' E) ∂environmentLaw μ := by
rw [freshPair,Measure.compProd_apply hE]
simp only [Kernel.prod_apply,freshPathKernel,Kernel.comap_apply]
lemma freshPair_first {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x z : Site d)
{E : Set (Environment d × Path d)} (hE : MeasurableSet E) :
freshPair μ x z {ξ | (ξ.1,ξ.2.1) ∈ E} = freshJoint μ x E := by
change freshPair μ x z ((fun ξ => (ξ.1,ξ.2.1)) ⁻¹' E) = _
rw [freshPair_apply μ x z (hE.preimage (measurable_fst.prodMk measurable_snd.fst)),
freshJoint_apply μ x hE]
apply lintegral_congr
intro ω
rw [show (Prod.mk ω ⁻¹' (fun ξ : Environment d × (Path d × Path d) => (ξ.1,ξ.2.1)) ⁻¹' E) =
(Prod.mk ω ⁻¹' E) ×ˢ Set.univ by ext p; simp]
rw [Measure.prod_prod,measure_univ,mul_one]
lemma freshPair_second {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x z : Site d)
{E : Set (Environment d × Path d)} (hE : MeasurableSet E) :
freshPair μ x z {ξ | (ξ.1,ξ.2.2) ∈ E} = freshJoint μ z E := by
change freshPair μ x z ((fun ξ => (ξ.1,ξ.2.2)) ⁻¹' E) = _
rw [freshPair_apply μ x z (hE.preimage (measurable_fst.prodMk measurable_snd.snd)),
freshJoint_apply μ z hE]
apply lintegral_congr
intro ω
rw [show (Prod.mk ω ⁻¹' (fun ξ : Environment d × (Path d × Path d) => (ξ.1,ξ.2.2)) ⁻¹' E) =
Set.univ ×ˢ (Prod.mk ω ⁻¹' E) by ext p; simp]
rw [Measure.prod_prod,measure_univ,one_mul]
lemma wordPairFamily_transfer {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(x z : Site d) (A : Set (Word d × Word d))
(hdis : Pairwise (fun a b : A => Disjoint (wordPairCylinder a.val) (wordPairCylinder b.val)))
(hrows : ∀ p ∈ A, Disjoint (wordDepartures p.1) (wordDepartures p.2)) :
((annealed μ x).prod (annealed μ z)) (⋃ p : A, wordPairCylinder p.val) =
freshPair μ x z {ξ | ξ.2 ∈ ⋃ p : A, wordPairCylinder p.val} := by
have hm := MeasurableSet.iUnion (fun p : A => measurableSet_wordPairCylinder p.val)
change ((annealed μ x).prod (annealed μ z)) (⋃ p : A, wordPairCylinder p.val) =
freshPair μ x z (Prod.snd ⁻¹' (⋃ p : A, wordPairCylinder p.val))
rw [freshPair_apply μ x z (hm.preimage measurable_snd),
measure_iUnion hdis (fun p => measurableSet_wordPairCylinder p.val)]
simp only [wordPairCylinder,Measure.prod_prod]
simp_rw [wordPair_transfer μ x z _ _ (hrows _ (Subtype.property _))]
rw [← lintegral_tsum]
· apply lintegral_congr
intro ω
change (∑' p : A, quenchedKernel d (ω,x) (wordCylinder p.val.1) *
quenchedKernel d (ω,z) (wordCylinder p.val.2)) =
((quenchedKernel d (ω,x)).prod (quenchedKernel d (ω,z))) (⋃ p : A, wordPairCylinder p.val)
rw [measure_iUnion hdis (fun p => measurableSet_wordPairCylinder p.val)]
apply tsum_congr
intro p
rw [wordPairCylinder,Measure.prod_prod]
· intro p
exact ((((quenchedKernel d).measurable_coe (measurableSet_pathCylinder _ _)).comp
(measurable_id.prodMk measurable_const)).mul
(((quenchedKernel d).measurable_coe (measurableSet_pathCylinder _ _)).comp
(measurable_id.prodMk measurable_const))).aemeasurable
lemma wordPairFamily_visit_bound {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(x z : Site d) (A : Set (Word d × Word d))
(hdis : Pairwise (fun a b : A => Disjoint (wordPairCylinder a.val) (wordPairCylinder b.val)))
(hrows : ∀ p ∈ A, Disjoint (wordDepartures p.1) (wordDepartures p.2))
(V W : Set (Environment d × Path d)) (hV : MeasurableSet V) (hW : MeasurableSet W)
(hcover : ∀ ξ : Environment d × (Path d × Path d),
(∃ p : A, ξ.2 ∈ wordPairCylinder p.val) → (ξ.1,ξ.2.1) ∈ V ∨ (ξ.1,ξ.2.2) ∈ W) :
((annealed μ x).prod (annealed μ z)) (⋃ p : A, wordPairCylinder p.val) ≤
freshJoint μ x V + freshJoint μ z W := by
rw [wordPairFamily_transfer μ x z A hdis hrows]
calc
_ ≤ freshPair μ x z ({ξ | (ξ.1,ξ.2.1) ∈ V} ∪ {ξ | (ξ.1,ξ.2.2) ∈ W}) :=
measure_mono (fun ξ hξ => hcover ξ (Set.mem_iUnion.mp hξ))
_ ≤ _ := (measure_union_le _ _).trans_eq (by rw [freshPair_first μ x z hV,freshPair_second μ x z hW])
def firstContactPairs {d : ℕ} (T R S K : Set (Site d)) : Set (Word d × Word d) :=
{p | firstHitWord T p.1 ∧ firstHitWord (wordDepartures p.1) p.2 ∧
(∃ i < p.1.1, wordPath p.1 i = wordEnd p.2 ∧
(∃ j ≤ i, wordPath p.1 j ∈ R) ∧ ∀ j ≤ i, wordPath p.1 j ∉ K) ∧
(∃ j ≤ p.2.1, wordPath p.2 j ∈ S) ∧ ∀ j ≤ p.2.1, wordPath p.2 j ∉ K}
lemma firstContactPairs_rows {d : ℕ} (T R S K : Set (Site d)) (p : Word d × Word d)
(hp : p ∈ firstContactPairs T R S K) :
Disjoint (wordDepartures p.1) (wordDepartures p.2) := by
apply Set.disjoint_left.mpr
rintro y hy ⟨j,hj,hjy⟩
exact hp.2.1.2 j hj (hjy ▸ hy)
lemma firstContactPairs_disjoint {d : ℕ} (T R S K : Set (Site d)) :
Pairwise (fun p q : firstContactPairs T R S K =>
Disjoint (wordPairCylinder p.val) (wordPairCylinder q.val)) := by
intro p q hpq
apply Set.disjoint_left.mpr
rintro ⟨X,Y⟩ ⟨hp1,hp2⟩ ⟨hq1,hq2⟩
by_cases h1 : p.val.1 = q.val.1
· have h2 : p.val.2 ≠ q.val.2 := by
intro h2
apply hpq
exact Subtype.ext (Prod.ext h1 h2)
have ha := p.property.2.1
have hb := q.property.2.1
rw [← h1] at hb
exact Set.disjoint_left.mp (firstHitWords_prefixFree (wordDepartures p.val.1)
(i := ⟨p.val.2,ha⟩) (j := ⟨q.val.2,hb⟩)
(fun h => h2 (congrArg Subtype.val h))) hp2 hq2
· exact Set.disjoint_left.mp (firstHitWords_prefixFree T
(i := ⟨p.val.1,p.property.1⟩) (j := ⟨q.val.1,q.property.1⟩)
(fun h => h1 (congrArg Subtype.val h))) hp1 hq1
lemma firstContactPairs_classify {d : ℕ} (T R S K : Set (Site d))
(A : Set (Site d)) (p : firstContactPairs T R S K) (X Y : Path d)
(h : (X,Y) ∈ wordPairCylinder p.val) :
X ∈ visitAfter R (A \ K) K ∨ Y ∈ visitAfter S (Aᶜ \ K) K := by
change (X ∈ wordCylinder p.val.1 ∧ Y ∈ wordCylinder p.val.2) at h
obtain ⟨i,hi,he,⟨j,hji,hR⟩,hK⟩ := p.property.2.2.1
have hyK : wordEnd p.val.2 ∉ K := by rw [← he];exact hK i le_rfl
by_cases hy : wordEnd p.val.2 ∈ A
· left
refine ⟨i,?_,⟨j,hji,?_⟩,fun k hk => ?_⟩
· rw [h.1 i hi.le,he]
exact ⟨hy,hyK⟩
· rw [h.1 j (hji.trans hi.le)]
exact hR
· rw [h.1 k (hk.trans hi.le)]
exact hK k hk
· right
obtain ⟨j,hj,hS⟩ := p.property.2.2.2.1
refine ⟨p.val.2.1,?_,⟨j,hj,?_⟩,fun k hk => ?_⟩
· rw [h.2 _ le_rfl]
exact ⟨hy,hyK⟩
· rw [h.2 j hj]
exact hS
· rw [h.2 k hk]
exact p.property.2.2.2.2 k hk
lemma firstContactPairs_visit_bound {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(x z : Site d) (T R S K : Set (Site d)) (A : Environment d → Set (Site d))
(hA : ∀ y, MeasurableSet {ω | y ∈ A ω}) :
((annealed μ x).prod (annealed μ z))
(⋃ p : firstContactPairs T R S K, wordPairCylinder p.val) ≤
freshJoint μ x {ξ | ξ.2 ∈ visitAfter R (A ξ.1 \ K) K} +
freshJoint μ z {ξ | ξ.2 ∈ visitAfter S ((A ξ.1)ᶜ \ K) K} := by
apply wordPairFamily_visit_bound μ x z _ (firstContactPairs_disjoint T R S K)
(firstContactPairs_rows T R S K) _ _
(measurable_random_visitAfter _ _ _ (fun y => (hA y).diff (MeasurableSet.const _)))
(measurable_random_visitAfter _ _ _ (fun y => (hA y).compl.diff (MeasurableSet.const _)))
rintro ⟨ω,X,Y⟩ ⟨p,hp⟩
exact firstContactPairs_classify T R S K (A ω) p X Y hp
lemma freshJoint_visitAfter_bound {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(x : Site d) (R B C : Set (Site d)) (A : Environment d → Set (Site d))
(hA : ∀ y, MeasurableSet {ω | y ∈ A ω}) (c : ℝ≥0∞)
(hc : ∀ᵐ ω ∂environmentLaw μ, ∀ y ∈ A ω,
c ≤ quenchedKernel d (ω,y) (hitBefore B C)) :
c * freshJoint μ x {ξ | ξ.2 ∈ visitAfter R (A ξ.1) (B ∪ C)} ≤
annealed μ x (hitBefore R B ∩ hitBefore B C) := by
rw [freshJoint_apply μ x (measurable_random_visitAfter _ _ _ hA),
annealed_apply μ x ((measurableSet_hitBefore _ _).inter (measurableSet_hitBefore _ _))]
apply (lintegral_const_mul_le _ _).trans
apply lintegral_mono_ae
filter_upwards [hc] with ω hω
exact quenched_visitAfter_bound ω x R (A ω) B C c hω
lemma firstContactPairs_raw_bound {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(x z : Site d) (T R S B C : Set (Site d))
(hex : ∀ᵐ ω ∂environmentLaw μ, ∀ y,
quenchedKernel d (ω,y) (hitBefore B C) + quenchedKernel d (ω,y) (hitBefore C B) = 1) :
((annealed μ x).prod (annealed μ z))
(⋃ p : firstContactPairs T R S (B ∪ C), wordPairCylinder p.val) ≤
2*(annealed μ x (hitBefore R B ∩ hitBefore B C) +
annealed μ z (hitBefore S C ∩ hitBefore C B)) := by
let A : Environment d → Set (Site d) := fun ω =>
{y | (2 : ℝ≥0∞)⁻¹ ≤ quenchedKernel d (ω,y) (hitBefore B C)}
have hA (y : Site d) : MeasurableSet {ω | y ∈ A ω} :=
measurableSet_le measurable_const (((quenchedKernel d).measurable_coe (measurableSet_hitBefore _ _)).comp
(measurable_id.prodMk measurable_const))
have hv := firstContactPairs_visit_bound μ x z T R S (B ∪ C) A hA
have h1 := freshJoint_visitAfter_bound μ x R B C (fun ω => A ω \ (B ∪ C))
(fun y => (hA y).diff (MeasurableSet.const _)) (2 : ℝ≥0∞)⁻¹ (by
filter_upwards [] with ω
exact fun y hy => hy.1)
have h2 := freshJoint_visitAfter_bound μ z S C B (fun ω => (A ω)ᶜ \ (B ∪ C))
(fun y => (hA y).compl.diff (MeasurableSet.const _)) (2 : ℝ≥0∞)⁻¹ (by
filter_upwards [hex] with ω hω
intro y hy
have hy' : quenchedKernel d (ω,y) (hitBefore B C) ≤ (2 : ℝ≥0∞)⁻¹ := le_of_lt (lt_of_not_ge hy.1)
apply (ENNReal.add_le_add_iff_left (by norm_num : (2 : ℝ≥0∞)⁻¹ ≠ ∞)).mp
calc
(2 : ℝ≥0∞)⁻¹ + (2 : ℝ≥0∞)⁻¹ = 1 := ENNReal.inv_two_add_inv_two
_ = _ := (hω y).symm
_ ≤ _ := add_le_add hy' le_rfl)
rw [Set.union_comm C B] at h2
have h : (2 : ℝ≥0∞)⁻¹ * ((annealed μ x).prod (annealed μ z))
(⋃ p : firstContactPairs T R S (B ∪ C), wordPairCylinder p.val) ≤
annealed μ x (hitBefore R B ∩ hitBefore B C) +
annealed μ z (hitBefore S C ∩ hitBefore C B) := by
calc
_ ≤ (2 : ℝ≥0∞)⁻¹ * (freshJoint μ x {ξ | ξ.2 ∈ visitAfter R (A ξ.1 \ (B ∪ C)) (B ∪ C)} +
freshJoint μ z {ξ | ξ.2 ∈ visitAfter S ((A ξ.1)ᶜ \ (B ∪ C)) (B ∪ C)}) := by gcongr
_ ≤ _ := by rw [mul_add]; exact add_le_add h1 h2
have h' := mul_le_mul (show (2 : ℝ≥0∞) ≤ 2 from le_rfl) h zero_le zero_le
simpa only [← mul_assoc,ENNReal.mul_inv_cancel (by norm_num : (2 : ℝ≥0∞) ≠ 0) (by norm_num : (2 : ℝ≥0∞) ≠ ∞),one_mul] using h'
end DirectionalZeroOne
end OAI
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