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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 | import OAI.Dynamics.ConditionalShuffle.PhysicalOne
namespace OAI
noncomputable section
namespace Revealed.Physical
open scoped Classical
open Thorp Thorp.Conditional Revealed.Split Revealed.Instrument
variable {ι α : Type} [fintype_ι : Fintype ι] [fintype_α : Fintype α] [decidableEq_α : DecidableEq α]
def instrument (d : ℕ) : Data (Outside ι α (Position d)) (Outside ι α (Position d))
(Equiv.Perm α) (Coins d) where
obs e c := nextOutside e (step d c)
inc e c := increment e (step d c)
next _ s := s
lemma physical_base (d : ℕ) (L : Sum ι α ≃ Position d) (t : ℕ) (ω : History d t) :
base (instrument d) (Split.outside L) t (observed (instrument d) (Split.outside L) t ω) =
Split.outside (L.trans (run d t ω)) := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
induction t with
| zero => simp only [base, run_zero]; rfl
| succ t ih =>
simp only [base, observed, Fin.init_snoc, Fin.snoc_last]
change nextOutside (base (instrument d) (Split.outside L) t
(observed (instrument d) (Split.outside L) t (Fin.init ω))) (step d (ω (Fin.last t))) = _
rw [ih, ← outside_post, run_succ]
rfl
lemma physical_group (d : ℕ) (L : Sum ι α ≃ Position d) (t : ℕ) (ω : History d t) :
groupRun (instrument d) (Split.outside L) t ω * assignment L =
assignment (L.trans (run d t ω)) := by
induction t with
| zero => simp only [groupRun, run_zero, one_mul]; rfl
| succ t ih =>
simp only [groupRun]
change (increment (base (instrument d) (Split.outside L) t
(observed (instrument d) (Split.outside L) t (Fin.init ω))) (step d (ω (Fin.last t))) *
groupRun (instrument d) (Split.outside L) t (Fin.init ω)) * assignment L = _
rw [mul_assoc, ih, physical_base, ← assignment_post, run_succ]
rfl
def fullPath {d t : ℕ} (e : Outside ι α (Position d))
(p : Fin t → Outside ι α (Position d)) : Fin (t+1) → ι → Position d :=
Fin.cons e.val (fun i => (p i).val)
lemma fullPath_injective {d t : ℕ} (e : Outside ι α (Position d)) :
Function.Injective (fullPath (t:=t) e) := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
intro p q h
funext i
apply Subtype.ext
exact (congrFun h i.succ)
lemma physical_path (d : ℕ) (L : Sum ι α ≃ Position d) (t : ℕ) (ω : History d t) :
fullPath (Split.outside L) (observed (instrument d) (Split.outside L) t ω) = splitPath L t ω := by
induction t with
| zero =>
funext i
exact Fin.cases rfl (fun j => Fin.elim0 j) i
| succ t ih =>
rw [show ω = Fin.snoc (Fin.init ω) (ω (Fin.last t)) from (Fin.snoc_init_self ω).symm]
simp only [observed, Fin.init_snoc, Fin.snoc_last, fullPath]
have he := Fin.comp_snoc (fun o : Outside ι α (Position d) => o.val)
(observed (instrument d) (Split.outside L) t (Fin.init ω))
((instrument d).obs (base (instrument d) (Split.outside L) t
(observed (instrument d) (Split.outside L) t (Fin.init ω))) (ω (Fin.last t)))
dsimp only [Function.comp_def] at he
rw [he, Fin.cons_snoc_eq_snoc_cons, ← fullPath, ih]
simp only [splitPath, exposedPath_snoc]
congr 1
rw [physical_base]
change (Split.outside (((sectionFrame (Split.outside (L.trans (run d t (Fin.init ω))))).trans
(step d (ω (Fin.last t)))))).val = _
funext i
change step d (ω (Fin.last t))
(sectionFrame (Split.outside (L.trans (run d t (Fin.init ω)))) (.inl i)) = _
rw [sectionFrame_spec]
rfl
lemma physical_group_frame (d : ℕ) (e : Outside ι α (Position d)) (t : ℕ) (ω : History d t) :
groupRun (instrument d) e t ω = assignment ((sectionFrame e).trans (run d t ω)) := by
simpa only [outside_sectionFrame, assignment_sectionFrame, mul_one] using
physical_group d (sectionFrame e) t ω
lemma physical_path_frame (d : ℕ) (e : Outside ι α (Position d)) (t : ℕ) (ω : History d t) :
fullPath e (observed (instrument d) e t ω) = splitPath (sectionFrame e) t ω := by
simpa only [outside_sectionFrame] using physical_path d (sectionFrame e) t ω
lemma physical_kernel_dichotomy [Nontrivial α] (d : ℕ)
(e s : Outside ι α (Position (d + 1))) :
Trim.signDichotomy (kernel (instrument (d + 1)) e s) :=
physical_step_signDichotomy d (sectionFrame e) s
end Revealed.Physical
end
end OAI
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