| 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 |
|
|
