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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 | import OAI.Dynamics.ConditionalShuffle.ResampledInstrument
namespace OAI
noncomputable section
open scoped Classical
namespace Revealed.Overlay
open Thorp Thorp.Conditional Revealed.Split Revealed.Disintegration Revealed.Instrument Revealed.Scheduled
variable {ι α : Type} [Fintype ι] [Fintype α] [DecidableEq α]
lemma step_relative_roundtrip_cons (d : ℕ) (c c' : Position d → Bool) (b : Bool) (x : Position d) :
step (d+1) c ((step (d+1) c').symm (step (d+1) c (Fin.cons b x))) =
step (d+1) c' (Fin.cons b x) := by
calc
_ = step (d+1) c ((step (d+1) c').symm (scatterPosition d (b ^^ c x,x))) := by
rw [step_cons]
_ = step (d+1) c (Fin.cons ((b ^^ c x) ^^ c' x) x) := by
erw [step_symm_scatter]; rfl
_ = scatterPosition d (((b ^^ c x) ^^ c' x) ^^ c x,x) := step_cons d c _ x
_ = scatterPosition d (b ^^ c' x,x) := by
apply congrArg (scatterPosition d)
apply Prod.ext
· cases b <;> cases c x <;> cases c' x <;> rfl
· rfl
_ = _ := (step_cons d c' b x).symm
lemma step_relative_roundtrip (d : ℕ) (c c' : Coins (d+1)) (x : Position (d+1)) :
step (d+1) c ((step (d+1) c').symm (step (d+1) c x)) = step (d+1) c' x := by
have h := step_relative_roundtrip_cons d c c' (x 0) (Fin.tail x)
simpa only [Fin.cons_self_tail] using h
lemma multiplier_forward (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) (a : α) :
step (d+1) c (L (.inr (multiplier d L active c r a))) =
step (d+1) (xorCoins d L active c r) (L (.inr a)) := by
have hm := relative_spec (L.trans (step (d+1) c))
(L.trans (step (d+1) (xorCoins d L active c r)))
(fun i => (xorCoins_occupied d L active c r i).symm) a
change step (d+1) (xorCoins d L active c r) (L (.inr (multiplier d L active c r a))) =
step (d+1) c (L (.inr a)) at hm
have he := congrArg (step (d+1) (xorCoins d L active c r)).symm hm
rw [Equiv.symm_apply_apply] at he
rw [he]
exact step_relative_roundtrip d c _ _
lemma multiplier_factor (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) :
(Equiv.sumCongr (Equiv.refl ι) (multiplier d L active c r)).trans (L.trans (step (d+1) c)) =
L.trans (step (d+1) (xorCoins d L active c r)) := by
apply Equiv.ext
intro x
cases x with
| inl i => exact (xorCoins_occupied d L active c r i).symm
| inr a => exact multiplier_forward d L active c r a
lemma multiplier_gauge (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) :
multiplier d L active c r = (assignment (L.trans (step (d+1) c)))⁻¹ *
increment (Split.outside L) (step (d+1) (xorCoins d L active c r)) * assignment L := by
have hh := congrArg assignment (multiplier_factor d L active c r)
rw [assignment_pre] at hh
rw [assignment_post L (step (d+1) (xorCoins d L active c r))] at hh
rw [mul_assoc]
exact (eq_inv_mul_iff_mul_eq).2 hh
def xorIncrement (d : ℕ) (σ : ℕ → Bool) (e : ℕ × Outside ι α (Position (d+1)))
(c r : Coins (d+1)) : Equiv.Perm α :=
increment e.2 (step (d+1) (xorCoins d (sectionFrame e.2) (σ e.1) c r))
lemma xorIncrement_law (d : ℕ) (σ : ℕ → Bool) (e : ℕ × Outside ι α (Position (d+1)))
(c : Coins (d+1)) : fairMass (xorIncrement d σ e c) =
kernel (stepInstrument (d+1) σ) e ((stepInstrument (d+1) σ).obs e c) := xor_kernel d σ e c
lemma xorCoins_frame (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) : xorCoins d L active c r =
xorCoins d (sectionFrame (Split.outside L)) active c r := by
rw [xor_resample, xor_resample]
unfold resample
rw [mask_frame]
lemma observed_base (d : ℕ) (L : Sum ι α ≃ Position d) (σ : ℕ → Bool) (t : ℕ) (c : History d t) :
base (stepInstrument d σ) (0,Split.outside L) t (observed (stepInstrument d σ) (0,Split.outside L) t c) =
(t,Split.outside (L.trans (run d t c))) := by
apply Prod.ext
· simpa only [Nat.zero_add] using base_clock d σ 0 (Split.outside L) t _
· rw [base_outside, observed_outside, Physical.physical_base]
lemma overlayRun_gauge (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (σ : ℕ → Bool)
(t : ℕ) (c r : History (d+1) t) :
overlayRun d L σ t c r = (assignment (L.trans (run (d+1) t c)))⁻¹ *
resampledRun (stepInstrument (d+1) σ) (xorIncrement d σ) (0,Split.outside L) t c r * assignment L := by
induction t with
| zero => simp only [overlayRun, run_zero, resampledRun, mul_one]; exact (inv_mul_cancel _).symm
| succ t ih =>
rw [overlayRun, multiplier_gauge, ih, resampledRun, observed_base]
rw [xorCoins_frame, run_succ]
change (assignment ((L.trans (run (d+1) t (Fin.init c))).trans (step (d+1) (c (Fin.last t)))))⁻¹ *
increment _ (step (d+1) (xorCoins d (sectionFrame (Split.outside (L.trans (run (d+1) t (Fin.init c)))))
(σ t) (c (Fin.last t)) (r (Fin.last t)))) * assignment (L.trans (run (d+1) t (Fin.init c))) *
((assignment (L.trans (run (d+1) t (Fin.init c))))⁻¹ * _ * assignment L) =
(assignment ((L.trans (run (d+1) t (Fin.init c))).trans (step (d+1) (c (Fin.last t)))))⁻¹ *
(increment _ (step (d+1) (xorCoins d (sectionFrame (Split.outside (L.trans (run (d+1) t (Fin.init c)))))
(σ t) (c (Fin.last t)) (r (Fin.last t)))) * _) * assignment L
group
end Revealed.Overlay
end
end OAI
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