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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace Problem367
@[reducible] def finiteSignalField {Ω ι : Type*} (m : ι → MeasurableSpace Ω) (I : Finset ι) :
MeasurableSpace Ω := ⨆ i ∈ I, m i
theorem finiteSignalField_le {Ω ι : Type*} {mΩ : MeasurableSpace Ω}
{m : ι → MeasurableSpace Ω} (hm : ∀ i, m i ≤ mΩ) (I : Finset ι) :
finiteSignalField m I ≤ mΩ := iSup_le fun i => iSup_le fun _ => hm i
@[simp] theorem finiteSignalField_singleton {Ω ι : Type*} [DecidableEq ι]
(m : ι → MeasurableSpace Ω) (i : ι) : finiteSignalField m {i} = m i := by
simp [finiteSignalField]
@[simp] theorem finiteSignalField_insert {Ω ι : Type*} [DecidableEq ι]
(m : ι → MeasurableSpace Ω) (I : Finset ι) (i : ι) :
finiteSignalField m (insert i I) = m i ⊔ finiteSignalField m I := by
simp [finiteSignalField, Finset.mem_insert, iSup_or, iSup_sup_eq]
theorem indep_finiteSignalField_of_not_mem {Ω ι : Type*} {mΩ : MeasurableSpace Ω}
{mu : Measure Ω} {m : ι → MeasurableSpace Ω} (hm : ∀ i, m i ≤ mΩ)
(hInd : iIndep m mu) {I : Finset ι} {a : ι} (ha : a ∉ I) :
Indep (finiteSignalField m I) (m a) mu := by
have hd : Disjoint (I : Set ι) ({a} : Set ι) := by
simpa only [Set.disjoint_singleton_right, Finset.mem_coe] using ha
simpa [finiteSignalField] using indep_iSup_of_disjoint hm hInd hd
theorem condExp_spin_sup_eq_binaryFusion
{Ω : Type*} {m₁ m₂ mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] (hm₁ : m₁ ≤ mΩ) (hm₂ : m₂ ≤ mΩ)
{S : Ω → IsingSpin} (hS : Measurable S)
(h₁ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₁] x| < 1)
(h₂ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₂] x| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, Indep m₁ m₂ (cond rho {x | S x = a})) :
rho[fun y => spinSign (S y) | m₁ ⊔ m₂] =ᵐ[rho]
fun x => binaryFusion (rho[fun y => spinSign (S y) | m₁] x)
(rho[fun y => spinSign (S y) | m₂] x) := by
have hX : @Measurable Ω Ω mΩ m₁ id := measurable_id.mono hm₁ le_rfl
have hY : @Measurable Ω Ω mΩ m₂ id := measurable_id.mono hm₂ le_rfl
have hq : @Measurable Ω ℝ m₁ inferInstance
(rho[fun y => spinSign (S y) | m₁]) := stronglyMeasurable_condExp.measurable
have hr : @Measurable Ω ℝ m₂ inferInstance
(rho[fun y => spinSign (S y) | m₂]) := stronglyMeasurable_condExp.measurable
have hI : ∀ a : IsingSpin,
@IndepFun Ω Ω Ω mΩ m₁ m₂ id id (cond rho {x | S x = a}) := by
intro a
simpa [IndepFun, Kernel.IndepFun, Indep] using hInd a
have h := @condExp_spin_given_observation_pair_eq_binaryFusion
Ω Ω Ω mΩ m₁ m₂ rho _ S hS id id hX hY
(rho[fun y => spinSign (S y) | m₁])
(rho[fun y => spinSign (S y) | m₂]) hq hr
(by simp) (by simp) h₁ h₂ hfair hI
change rho[fun y => spinSign (S y) |
(m₁.prod m₂).comap (fun x : Ω => (id x, id x))] =ᵐ[rho] _ at h
rw [MeasurableSpace.comap_prodMk, MeasurableSpace.comap_id,
MeasurableSpace.comap_id] at h
exact h
theorem abs_condExp_spin_sup_lt_one
{Ω : Type*} {m₁ m₂ mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] (hm₁ : m₁ ≤ mΩ) (hm₂ : m₂ ≤ mΩ)
{S : Ω → IsingSpin} (hS : Measurable S)
(h₁ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₁] x| < 1)
(h₂ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₂] x| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, Indep m₁ m₂ (cond rho {x | S x = a})) :
∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₁ ⊔ m₂] x| < 1 := by
filter_upwards [condExp_spin_sup_eq_binaryFusion hm₁ hm₂ hS h₁ h₂ hfair hInd,
h₁, h₂] with x hx hx₁ hx₂
rw [hx]
exact abs_binaryFusion_lt_one hx₁ hx₂
theorem abs_condExp_spin_finiteSignalField_lt_one
{Ω ι : Type*} {mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] {m : ι → MeasurableSpace Ω}
(hm : ∀ i, m i ≤ mΩ) {S : Ω → IsingSpin} (hS : Measurable S)
(hstrict : ∀ i, ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m i] x| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, iIndep m (cond rho {x | S x = a}))
(I : Finset ι) (hI : I.Nonempty) :
∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | finiteSignalField m I] x| < 1 := by
classical
induction I using Finset.induction_on with
| empty => simp at hI
| @insert a I ha ih =>
by_cases hIn : I.Nonempty
· rw [finiteSignalField_insert, sup_comm]
exact abs_condExp_spin_sup_lt_one (finiteSignalField_le hm I) (hm a) hS
(ih hIn) (hstrict a) hfair
(fun b => indep_finiteSignalField_of_not_mem hm (hInd b) ha)
· have hIe : I = ∅ := Finset.not_nonempty_iff_eq_empty.mp hIn
subst I
simpa using hstrict a
def spinInformation {Ω : Type*} {mΩ : MeasurableSpace Ω}
(rho : Measure Ω) (S : Ω → IsingSpin) (m : MeasurableSpace Ω) : ℝ :=
∫ x, (rho[fun y => spinSign (S y) | m] x) ^ 2 ∂rho
theorem spinInformation_sup_le_add
{Ω : Type*} {m₁ m₂ mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] (hm₁ : m₁ ≤ mΩ) (hm₂ : m₂ ≤ mΩ)
{S : Ω → IsingSpin} (hS : Measurable S)
(h₁ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₁] x| < 1)
(h₂ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₂] x| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, Indep m₁ m₂ (cond rho {x | S x = a}))
(hsym : Measure.map (fun u : ℝ => -u)
(Measure.map (rho[fun y => spinSign (S y) | m₂]) rho) =
Measure.map (rho[fun y => spinSign (S y) | m₂]) rho) :
spinInformation rho S (m₁ ⊔ m₂) ≤
spinInformation rho S m₁ + spinInformation rho S m₂ := by
have hX : @Measurable Ω Ω mΩ m₁ id := measurable_id.mono hm₁ le_rfl
have hY : @Measurable Ω Ω mΩ m₂ id := measurable_id.mono hm₂ le_rfl
have hI : ∀ a : IsingSpin,
@IndepFun Ω Ω Ω mΩ m₁ m₂ id id (cond rho {x | S x = a}) := by
intro a
simpa [IndepFun, Kernel.IndepFun, Indep] using hInd a
have h := @condExp_observation_pair_secondMoment_le_add
Ω Ω Ω mΩ m₁ m₂ rho _ S hS id id hX hY
(rho[fun y => spinSign (S y) | m₁])
(rho[fun y => spinSign (S y) | m₂])
stronglyMeasurable_condExp.measurable stronglyMeasurable_condExp.measurable
(by simp) (by simp) h₁ h₂ hfair hI hsym
change spinInformation rho S
((m₁.prod m₂).comap (fun x : Ω => (id x, id x))) ≤ _ at h
rw [MeasurableSpace.comap_prodMk, MeasurableSpace.comap_id,
MeasurableSpace.comap_id] at h
exact h
theorem spinInformation_sup_le_sub_loss
{Ω : Type*} {m₁ m₂ mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] (hm₁ : m₁ ≤ mΩ) (hm₂ : m₂ ≤ mΩ)
{S : Ω → IsingSpin} (hS : Measurable S)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, Indep m₁ m₂ (cond rho {x | S x = a}))
{theta : ℝ}
(h₁ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₁] x| ≤ theta)
(h₂ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₂] x| ≤ theta)
(htheta : theta < 1)
(hsym : Measure.map (fun u : ℝ => -u)
(Measure.map (rho[fun y => spinSign (S y) | m₂]) rho) =
Measure.map (rho[fun y => spinSign (S y) | m₂]) rho) :
spinInformation rho S (m₁ ⊔ m₂) ≤
spinInformation rho S m₁ + spinInformation rho S m₂ -
2 * (1 - theta ^ 2) * spinInformation rho S m₁ * spinInformation rho S m₂ := by
have hX : @Measurable Ω Ω mΩ m₁ id := measurable_id.mono hm₁ le_rfl
have hY : @Measurable Ω Ω mΩ m₂ id := measurable_id.mono hm₂ le_rfl
have hI : ∀ a : IsingSpin,
@IndepFun Ω Ω Ω mΩ m₁ m₂ id id (cond rho {x | S x = a}) := by
intro a
simpa [IndepFun, Kernel.IndepFun, Indep] using hInd a
have h := @condExp_observation_pair_secondMoment_le
Ω Ω Ω mΩ m₁ m₂ rho _ S hS id id hX hY
(rho[fun y => spinSign (S y) | m₁])
(rho[fun y => spinSign (S y) | m₂])
stronglyMeasurable_condExp.measurable stronglyMeasurable_condExp.measurable
(by simp) (by simp) hfair hI theta h₁ h₂ htheta hsym
change spinInformation rho S
((m₁.prod m₂).comap (fun x : Ω => (id x, id x))) ≤ _ at h
rw [MeasurableSpace.comap_prodMk, MeasurableSpace.comap_id,
MeasurableSpace.comap_id] at h
exact h
section FiniteFamily
variable {Ω ι : Type*} {mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] {m : ι → MeasurableSpace Ω}
(hm : ∀ i, m i ≤ mΩ) {S : Ω → IsingSpin} (hS : Measurable S)
(hstrict : ∀ i, ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m i] x| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, iIndep m (cond rho {x | S x = a}))
(hsym : ∀ i, Measure.map (fun u : ℝ => -u)
(Measure.map (rho[fun y => spinSign (S y) | m i]) rho) =
Measure.map (rho[fun y => spinSign (S y) | m i]) rho)
include hm hS hstrict hfair hInd hsym
theorem spinInformation_finiteSignalField_union_le [DecidableEq ι]
(J : Finset ι) (hJ : J.Nonempty) (I : Finset ι) (hIJ : Disjoint I J) :
spinInformation rho S (finiteSignalField m (I ∪ J)) ≤
(∑ i ∈ I, spinInformation rho S (m i)) + spinInformation rho S (finiteSignalField m J) := by
induction I using Finset.induction_on with
| empty => simp
| @insert a I ha ih =>
have hI : Disjoint I J := (Finset.disjoint_insert_left.mp hIJ).2
have haJ : a ∉ J := (Finset.disjoint_insert_left.mp hIJ).1
have haIJ : a ∉ I ∪ J := by simp [ha, haJ]
have hne : (I ∪ J).Nonempty := hJ.mono Finset.subset_union_right
have hbound := spinInformation_sup_le_add (finiteSignalField_le hm (I ∪ J))
(hm a) hS
(abs_condExp_spin_finiteSignalField_lt_one hm hS hstrict hfair hInd (I ∪ J) hne)
(hstrict a) hfair
(fun b => indep_finiteSignalField_of_not_mem hm (hInd b) haIJ) (hsym a)
have hih := ih hI
rw [Finset.insert_union, finiteSignalField_insert, sup_comm]
rw [Finset.sum_insert ha]
linarith
theorem spinInformation_finiteSignalField_le_sum
(I : Finset ι) (hI : I.Nonempty) :
spinInformation rho S (finiteSignalField m I) ≤
∑ i ∈ I, spinInformation rho S (m i) := by
classical
induction I using Finset.induction_on with
| empty => simp at hI
| @insert a I ha ih =>
by_cases hIn : I.Nonempty
· have hbound := spinInformation_sup_le_add (finiteSignalField_le hm I) (hm a) hS
(abs_condExp_spin_finiteSignalField_lt_one hm hS hstrict hfair hInd I hIn)
(hstrict a) hfair
(fun b => indep_finiteSignalField_of_not_mem hm (hInd b) ha) (hsym a)
rw [finiteSignalField_insert, sup_comm, Finset.sum_insert ha]
linarith [ih hIn]
· have hIe : I = ∅ := Finset.not_nonempty_iff_eq_empty.mp hIn
subst I
simp
theorem spinInformation_finiteSignalField_le_sum_sub_loss
(I : Finset ι) {a b : ι} (hab : a ≠ b) (ha : a ∈ I) (hb : b ∈ I)
{theta : ℝ}
(haTheta : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m a] x| ≤ theta)
(hbTheta : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m b] x| ≤ theta)
(htheta : theta < 1) :
spinInformation rho S (finiteSignalField m I) ≤
(∑ i ∈ I, spinInformation rho S (m i)) -
2 * (1 - theta ^ 2) * spinInformation rho S (m a) * spinInformation rho S (m b) := by
classical
have hJsub : ({a, b} : Finset ι) ⊆ I := by
intro c hc
rcases Finset.mem_insert.mp hc with rfl | hc
· exact ha
· have hc' : c = b := Finset.mem_singleton.mp hc
simpa only [hc'] using hb
have hdisj : Disjoint (I \ {a, b}) ({a, b} : Finset ι) := by
apply Finset.disjoint_left.mpr
intro c hc hcJ
exact (Finset.mem_sdiff.mp hc).2 hcJ
have hunion := spinInformation_finiteSignalField_union_le hm hS hstrict hfair hInd hsym
{a, b} (by simp) (I \ {a, b}) hdisj
rw [Finset.sdiff_union_of_subset hJsub] at hunion
have hpairInd (s : IsingSpin) : Indep (m a) (m b) (cond rho {x | S x = s}) := by
simpa using indep_finiteSignalField_of_not_mem hm (hInd s)
(I := {a}) (a := b) (by simp [Ne.symm hab])
have hpair := spinInformation_sup_le_sub_loss (hm a) (hm b) hS hfair hpairInd
haTheta hbTheta htheta (hsym b)
have hsum := Finset.sum_sdiff (f := fun i => spinInformation rho S (m i)) hJsub
simp only [finiteSignalField_insert, finiteSignalField_singleton] at hunion
simp [hab] at hsum
linarith
end FiniteFamily
end Problem367
end
end OAI
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