File size: 4,845 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Dynamics.ConditionalShuffle.TVTransport
namespace OAI
noncomputable section
namespace Revealed.Split
open scoped Classical
def slotEquiv {α β : Type} (r₀ r₁ : α ≃ β) (a₀ : Equiv.Perm α) : Equiv.Perm α ≃ Equiv.Perm β :=
r₁.permCongr.trans (Equiv.mulRight (r₀.symm.trans (a₀.trans r₁)))
lemma slotEquiv_apply {α β : Type} (r₀ r₁ : α ≃ β) (a₀ g : Equiv.Perm α) :
slotEquiv r₀ r₁ a₀ g = r₀.symm.trans ((a₀.trans g).trans r₁) := by
apply Equiv.ext
intro x
change r₁ (g (r₁.symm (r₁ (a₀ (r₀.symm x))))) = r₁ (g (a₀ (r₀.symm x)))
rw [Equiv.symm_apply_apply]
end Revealed.Split
namespace Revealed.Disintegration
open scoped Classical
open Thorp
lemma average_quotient {E G : Type} [Fintype E] [Fintype G] [Nonempty G]
(μ : E × G → ℝ) (u : G → ℝ) :
(∑ e, marginal μ e * tv (fun g => μ (e,g) / marginal μ e) u) =
∑ e, marginal μ e * tv (conditional μ e) u := by
apply Finset.sum_congr rfl
intro e _
by_cases h : marginal μ e = 0
· simp only [h, zero_mul]
· have hc : conditional μ e = fun g => μ (e,g) / marginal μ e := by
funext g
rw [conditional, ite_eq_right h]
rw [hc]
end Revealed.Disintegration
namespace Revealed
open scoped Classical
open Thorp Thorp.Conditional Split Instrument Disintegration Physical
def splitLayout (d : ℕ) (g₀ : State (d+1)) : Sum (Position d) (Position d) ≃ Position (d+1) :=
(splitLabels d).trans g₀
lemma splitLayout_post (d : ℕ) (g₀ g : State (d+1)) :
(splitLayout d g₀).trans g = splitLayout d (g*g₀) := rfl
lemma outsidePaths_split (d t : ℕ) (g₀ : State (d+1)) (ω : History (d+1) t) :
outsidePaths d t g₀ ω = splitPath (splitLayout d g₀) t ω := by
induction t with
| zero => rfl
| succ t ih =>
funext s x
refine Fin.lastCases ?_ (fun j => ?_) s
· simp only [outsidePaths, layoutPath, splitPath, exposedPath, Fin.snoc_last]
rfl
· simp only [outsidePaths, layoutPath, splitPath, exposedPath, Fin.snoc_castSucc]
exact congrFun (congrFun (ih (Fin.init ω)) j) x
def canonicalTransport (d t : ℕ) (g₀ : State (d+1))
(p : Fin t → Outside (Position d) (Position d) (Position (d+1))) :
State d ≃ Equiv.Perm (Fin (2^d)) :=
slotEquiv (ranks d (splitLayout d g₀))
(ranks d (sectionFrame (base (instrument (d+1)) (Split.outside (splitLayout d g₀)) t p)))
(assignment (splitLayout d g₀))
lemma canonical_realization (d t : ℕ) (g₀ : State (d+1)) (ω : History (d+1) t) :
canonicalTransport d t g₀ (observed (instrument (d+1)) (Split.outside (splitLayout d g₀)) t ω)
(groupRun (instrument (d+1)) (Split.outside (splitLayout d g₀)) t ω) =
slotPermutation d t g₀ ω := by
rw [canonicalTransport, slotEquiv_apply, physical_base]
have hg : (assignment (splitLayout d g₀)).trans
(groupRun (instrument (d+1)) (Split.outside (splitLayout d g₀)) t ω) =
assignment ((splitLayout d g₀).trans (run (d+1) t ω)) :=
physical_group (d+1) (splitLayout d g₀) t ω
rw [hg, ← ranks_assignment, splitLayout_post]
change (ranks d ((splitLabels d).trans g₀)).symm.trans
(ranks d ((splitLabels d).trans (run (d+1) t ω*g₀))) = _
rw [ranks_freeRank, ranks_freeRank]
rfl
lemma expectedTV_completed (d t : ℕ) (g₀ : State (d+1)) : expectedTV d t g₀ =
∑ e, marginal (fairMass (fun ω : History (d+1) t =>
(outsidePaths d t g₀ ω,slotPermutation d t g₀ ω))) e *
tv (conditional (fairMass (fun ω : History (d+1) t =>
(outsidePaths d t g₀ ω,slotPermutation d t g₀ ω))) e)
(fun _ => (Fintype.card (Equiv.Perm (Fin (2^d))) : ℝ)⁻¹) := by
rw [← average_quotient]
simp only [marginal_fairMass, Fintype.card_perm, Fintype.card_fin]
simp only [expectedTV, environmentMass, one_div]
apply Finset.sum_congr rfl
intro e _
congr 2
theorem expectedTV_le_instrument (d t : ℕ) (g₀ : State (d+1)) : expectedTV d t g₀ ≤
Instrument.distance (instrument (d+1)) (Split.outside (splitLayout d g₀)) t := by
let : Fintype (Outside (Position d) (Position d) (Position (d+1))) := inferInstance
rw [expectedTV_completed, distance_disintegration]
have h := average_tv_transport
(observed (instrument (d+1)) (Split.outside (splitLayout d g₀)) t)
(groupRun (instrument (d+1)) (Split.outside (splitLayout d g₀)) t)
(fullPath (Split.outside (splitLayout d g₀))) (canonicalTransport d t g₀)
simp_rw [canonical_realization, physical_path, ← outsidePaths_split] at h
exact h
lemma expectedTV_nonneg (d t : ℕ) (g₀ : State (d+1)) : 0 ≤ expectedTV d t g₀ := by
unfold expectedTV
exact Finset.sum_nonneg (fun e _ => mul_nonneg (fairMass_nonneg _ e) (Thorp.Fourier.tv_nonneg _ _))
end Revealed
end
end OAI
|