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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import OAI.Dynamics.ConditionalShuffle.ResetMinorization
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace Revealed.Overlay
open Thorp Thorp.Conditional Revealed.Split Revealed.Disintegration
def maskedXor (d : ℕ) (m : Position d → Bool) (c r : Coins (d+1)) : Coins (d+1) :=
fun x => c x ^^ (if m x then r x else false)
lemma maskedXor_involutive (d : ℕ) (m : Position d → Bool) (r : Coins (d+1)) :
Function.Involutive (fun c => maskedXor d m c r) := by
intro c; funext x
change ((c x ^^ (if m x then r x else false)) ^^ (if m x then r x else false)) = c x
simp
def predictableHistory (d : ℕ)
(M : (t : ℕ) → History (d+1) t → History (d+1) t → Position d → Bool) :
(t : ℕ) → History (d+1) t → History (d+1) t → History (d+1) t
| 0, _, _ => Fin.elim0
| t+1, c, r => Fin.snoc (predictableHistory d M t (Fin.init c) (Fin.init r))
(maskedXor d (M t (Fin.init c) (Fin.init r)) (c (Fin.last t)) (r (Fin.last t)))
lemma predictableHistory_injective (d : ℕ)
(M : (t : ℕ) → History (d+1) t → History (d+1) t → Position d → Bool)
(t : ℕ) (r : History (d+1) t) : Function.Injective (fun c => predictableHistory d M t c r) := by
induction t with
| zero => intro c c' _; exact Subsingleton.elim _ _
| succ t ih =>
intro c c' h
have hp := congrArg Fin.init h
simp only [predictableHistory, Fin.init_snoc] at hp
have hc := ih (Fin.init r) hp
have hl := congrFun h (Fin.last t)
simp only [predictableHistory, Fin.snoc_last, hc] at hl
have hl' := (maskedXor_involutive d (M t (Fin.init c') (Fin.init r)) (r (Fin.last t))).injective hl
calc
c = Fin.snoc (Fin.init c) (c (Fin.last t)) := (Fin.snoc_init_self c).symm
_ = Fin.snoc (Fin.init c') (c' (Fin.last t)) := by rw [hc, hl']
_ = c' := Fin.snoc_init_self c'
def predictableEquiv (d : ℕ)
(M : (t : ℕ) → History (d+1) t → History (d+1) t → Position d → Bool)
(t : ℕ) (r : History (d+1) t) : Equiv.Perm (History (d+1) t) :=
Equiv.ofBijective (fun c => predictableHistory d M t c r)
((Fintype.bijective_iff_injective_and_card _).mpr
⟨predictableHistory_injective d M t r, rfl⟩)
lemma predictable_law (d : ℕ)
(M : (t : ℕ) → History (d+1) t → History (d+1) t → Position d → Bool)
(t : ℕ) (r : History (d+1) t) :
fairMass (fun c : History (d+1) t => run (d+1) t (predictableHistory d M t c r)) = law (d+1) t := by
exact fairMass_comp_equiv (run (d+1) t) (predictableEquiv d M t r)
variable {ι α : Type} [fintype_ι : Fintype ι] [fintype_α : Fintype α] [decidableEq_α : DecidableEq α]
def modifiedHistory (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool)
(t : ℕ) (c r : History (d+1) t) : History (d+1) t :=
predictableHistory d (fun k a _ => mask d (L.trans (run (d+1) k a)) (σ k)) t c r
lemma modifiedHistory_succ (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool)
(t : ℕ) (c r : History (d+1) (t+1)) :
modifiedHistory d L σ (t+1) c r =
Fin.snoc (modifiedHistory d L σ t (Fin.init c) (Fin.init r))
(xorCoins d (L.trans (run (d+1) t (Fin.init c))) (σ t) (c (Fin.last t)) (r (Fin.last t))) := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
exact rfl
theorem overlay_physical_law (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool)
(t : ℕ) (r : History (d+1) t) :
fairMass (fun c : History (d+1) t => run (d+1) t (modifiedHistory d L σ t c r)) = law (d+1) t := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
exact predictable_law d _ t r
lemma modified_outside (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool)
(t : ℕ) (c r : History (d+1) t) (i : ι) :
run (d+1) t (modifiedHistory d L σ t c r) (L (.inl i)) =
run (d+1) t c (L (.inl i)) := by
induction t with
| zero => simp only [run_zero, Equiv.Perm.one_apply]
| succ t ih =>
rw [modifiedHistory_succ, run_succ, run_succ]
simp only [Fin.snoc_last, Fin.snoc_castSucc, Equiv.Perm.mul_apply]
rw [ih]
exact xorCoins_occupied d (L.trans (run (d+1) t (Fin.init c))) (σ t) _ _ i
lemma modified_inside (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool)
(t : ℕ) (c r : History (d+1) t) (a : α) :
run (d+1) t (modifiedHistory d L σ t c r) (L (.inr a)) =
run (d+1) t c (L (.inr (overlayRun d L σ t c r a))) := by
induction t with
| zero => simp only [run_zero, overlayRun, Equiv.Perm.one_apply]
| succ t ih =>
rw [modifiedHistory_succ, run_succ, run_succ, overlayRun]
simp only [Fin.snoc_last, Fin.snoc_castSucc, Equiv.Perm.mul_apply]
rw [ih]
exact (multiplier_forward d (L.trans (run (d+1) t (Fin.init c))) (σ t) _ _ _).symm
theorem overlay_incoming_factor (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool)
(t : ℕ) (c r : History (d+1) t) (π : Equiv.Perm α) :
(Equiv.sumCongr (Equiv.refl ι) π).trans (L.trans (run (d+1) t (modifiedHistory d L σ t c r))) =
(Equiv.sumCongr (Equiv.refl ι) (overlayRun d L σ t c r * π)).trans (L.trans (run (d+1) t c)) := by
apply Equiv.ext; intro x
cases x with
| inl i => exact modified_outside d L σ t c r i
| inr a => exact modified_inside d L σ t c r (π a)
end Revealed.Overlay
end
end OAI
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