| import OAI.Dynamics.ConditionalShuffle.ResampleFiber |
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| namespace OAI |
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| 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] |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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] |
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| 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 |
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| 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 |
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| 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) |
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| 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 |
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| 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] |
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| 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 |
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| end Revealed.Overlay |
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| end |
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| end OAI |
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