| import OAI.Dynamics.ConditionalShuffle.SelectedSchedule |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped Classical |
| namespace Revealed.Instrument |
| open Thorp Thorp.Conditional |
| lemma Data.ext {E S G Ω : Type} {I J : Data E S G Ω} |
| (ho : I.obs = J.obs) (hi : I.inc = J.inc) (hn : I.next = J.next) : I = J := by |
| cases I |
| cases J |
| cases ho |
| cases hi |
| cases hn |
| rfl |
|
|
| variable {E E' S G Ω : Type} [fintype_S : Fintype S] [fintype_G : Fintype G] [group_G : Group G] [fintype_Ω : Fintype Ω] [nonempty_Ω : Nonempty Ω] |
| variable (I : Data E S G Ω) (J : Data E' S G Ω) (f : E → E') |
| (ho : ∀ e c, J.obs (f e) c = I.obs e c) |
| (hi : ∀ e c, J.inc (f e) c = I.inc e c) |
| (hn : ∀ e s, J.next (f e) s = f (I.next e s)) |
|
|
| include hn |
| lemma map_base (e : E) (n : ℕ) (p : Fin n → S) : |
| base J (f e) n p = f (base I e n p) := by |
| let retained_fintype_S := fintype_S |
| let retained_fintype_G := fintype_G |
| let retained_group_G := group_G |
| let retained_fintype_Ω := fintype_Ω |
| let retained_nonempty_Ω := nonempty_Ω |
| induction n with |
| | zero => rfl |
| | succ n ih => rw [base, ih, hn]; rfl |
|
|
| include ho |
| lemma map_observed (e : E) (n : ℕ) (c : Fin n → Ω) : |
| observed J (f e) n c = observed I e n c := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| simp only [observed, ih, map_base I J f hn, ho] |
|
|
| include hi |
| lemma map_groupRun (e : E) (n : ℕ) (c : Fin n → Ω) : |
| groupRun J (f e) n c = groupRun I e n c := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| simp only [groupRun, map_observed I J f ho hn, map_base I J f hn, hi, ih] |
|
|
| lemma map_distance (e : E) (n : ℕ) : distance J (f e) n = distance I e n := by |
| rw [distance_disintegration, distance_disintegration] |
| have hh : pathJoint J (f e) n = pathJoint I e n := by |
| unfold pathJoint |
| simp_rw [map_observed I J f ho hn, map_groupRun I J f ho hi hn] |
| rw [hh] |
|
|
| end Revealed.Instrument |
|
|
| end |
|
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| end OAI |
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|
