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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 | import OAI.Dynamics.ConditionalShuffle.ResampleFiber
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 ι] [fintype_α : Fintype α] [decidableEq_α : DecidableEq α]
def mask (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool) : Position d → Bool :=
fun x => active && freePairMask d (avoid (fun i => L (.inl i))) x
def resample (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) : Coins (d+1) := maskedCoin (mask d L active) c r
lemma resample_false (d : ℕ) (L : Sum ι α ≃ Position (d+1))
(c r : Coins (d+1)) : resample d L false c r = c := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
funext x
change (if false && freePairMask d (avoid (fun i => L (.inl i))) x then r x else c x) = c x
simp
lemma resample_occupied (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) (i : ι) :
step (d+1) (resample d L active c r) (L (.inl i)) = step (d+1) c (L (.inl i)) := by
cases active with
| false => rw [resample_false]
| true =>
apply masked_step_occupied
exact avoid_exposed _ i
lemma mask_false_witness (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (x : Position d)
(hx : freePairMask d (avoid (fun i => L (.inl i))) x = false) :
∃ i : ι, Fin.tail (L (.inl i)) = x := by
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
by_contra h
have hh : ∀ b : Bool, avoid (fun i => L (.inl i)) (Fin.cons b x) = true := by
intro b
simp only [avoid, decide_eq_true_eq]
intro i hi
apply h
exact ⟨i, by rw [hi]; rfl⟩
simp only [freePairMask, hh, Bool.and_self, Bool.true_eq_false] at hx
lemma resample_obs (d : ℕ) (σ : ℕ → Bool) (e : ℕ × Outside ι α (Position (d+1)))
(c r : Coins (d+1)) :
(stepInstrument (d+1) σ).obs e (resample d (sectionFrame e.2) (σ e.1) c r) =
(stepInstrument (d+1) σ).obs e c := by
apply Prod.ext
· apply Subtype.ext; funext i
exact resample_occupied d (sectionFrame e.2) (σ e.1) c r i
· cases h : σ e.1 <;> simp only [stepInstrument, h, Bool.false_eq_true, ite_false, ite_true, resample_false]
lemma resample_row (d : ℕ) (σ : ℕ → Bool) (e : ℕ × Outside ι α (Position (d+1)))
(c c' : Coins (d+1))
(hh : (stepInstrument (d+1) σ).obs e c = (stepInstrument (d+1) σ).obs e c')
(r : Coins (d+1)) :
resample d (sectionFrame e.2) (σ e.1) c r = resample d (sectionFrame e.2) (σ e.1) c' r := by
cases hs : σ e.1 with
| false =>
have hc := congrArg Prod.snd hh
have hc' : c = c' := by simpa only [stepInstrument, hs, Bool.false_eq_true, ite_false, Option.some.injEq] using hc
rw [hc']
| true =>
funext x
change (if mask d (sectionFrame e.2) true x then r x else c x) =
(if mask d (sectionFrame e.2) true x then r x else c' x)
simp only [mask, Bool.true_and]
cases hm : freePairMask d (avoid (fun i => sectionFrame e.2 (.inl i))) x with
| true => rfl
| false =>
obtain ⟨i,hi⟩ := mask_false_witness d (sectionFrame e.2) x hm
have ho := congrFun (congrArg (fun z => z.1.val) hh) i
have hc := (step_same_at_iff d c c' (sectionFrame e.2 (.inl i))).mp ho
simpa only [Bool.false_eq_true, ite_false, hi] using hc
lemma resample_kernel (d : ℕ) (σ : ℕ → Bool) (e : ℕ × Outside ι α (Position (d+1)))
(c : Coins (d+1)) :
fairMass (fun r : Coins (d+1) => (stepInstrument (d+1) σ).inc e
(resample d (sectionFrame e.2) (σ e.1) c r)) =
kernel (stepInstrument (d+1) σ) e ((stepInstrument (d+1) σ).obs e c) := by
apply resample_conditional _ _ _ (resample_obs d σ e) _ (resample_row d σ e) c
intro f
exact mean_maskedCoin _ f
def xorEquiv (d : ℕ) (c : Coins (d+1)) : Equiv.Perm (Coins (d+1)) :=
Function.Involutive.toPerm (fun r x => c x ^^ r x) (by
intro r; funext x
change (c x ^^ (c x ^^ r x)) = r x
cases c x <;> cases r x <;> rfl)
lemma xor_resample (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool)
(c r : Coins (d+1)) :
xorCoins d L active c r = resample d L active c (xorEquiv d c r) := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
funext x
change (c x ^^ (if mask d L active x then r x else false)) =
(if mask d L active x then c x ^^ r x else c x)
cases mask d L active x <;> simp
lemma mask_frame (d : ℕ) (L : Sum ι α ≃ Position (d+1)) (active : Bool) :
mask d L active = mask d (sectionFrame (Split.outside L)) active := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
have hh : (fun i => L (.inl i)) = (fun i => sectionFrame (Split.outside L) (.inl i)) := by
funext i; exact (sectionFrame_spec (Split.outside L) i).symm
unfold mask; rw [hh]
lemma xor_kernel (d : ℕ) (σ : ℕ → Bool) (e : ℕ × Outside ι α (Position (d+1)))
(c : Coins (d+1)) :
fairMass (fun r : Coins (d+1) => (stepInstrument (d+1) σ).inc e
(xorCoins d (sectionFrame e.2) (σ e.1) c r)) =
kernel (stepInstrument (d+1) σ) e ((stepInstrument (d+1) σ).obs e c) := by
have hh := resample_kernel d σ e c
refine Eq.trans ?_ hh
funext g
rw [fairMass_eq_mean, fairMass_eq_mean]
have hx := mean_equiv (xorEquiv d c)
(fun r => if (stepInstrument (d+1) σ).inc e (resample d (sectionFrame e.2) (σ e.1) c r) = g then (1 : ℝ) else 0)
convert hx using 1
apply mean_congr; intro r
rw [xor_resample]
rfl
end Revealed.Overlay
end
end OAI
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