File size: 3,006 Bytes
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 | import OAI.Computability.UniqueGames.Foundations.SamplingLemmas
import OAI.Computability.UniqueGames.Soundness.ZeroInformationLemmas
namespace OAI
section
/-! Explicit hidden-coordinate enumeration in the actual conditional law. -/
namespace UniqueGamesTheorem.Soundness.ConditionalIncidences
open scoped BigOperators
def coordinateReindex {A B X : Type} (e : B ≃ A) : (A → X) ≃ (B → X) where
toFun f := fun b => f (e b)
invFun g := fun a => g (e.symm a)
left_inv f := by
funext a
exact congrArg f (e.apply_symm_apply a)
right_inv g := by
funext b
exact congrArg g (e.symm_apply_apply b)
theorem raw_conditional_incidence_reindex
{P D O N : Type} [Fintype P] [DecidableEq P]
[Fintype D] [Zero D] [DecidableEq D]
[Fintype O] [DecidableEq O] [DecidableEq N]
(J : Finset P) (name : O → Fin 3 → N) (gamma : RawCoefficients J D)
(observed : PositionOutside (zeroSet J gamma) → O × N)
(hpos : 0 < Fintype.card (RawObservationFibre J name gamma observed))
(r : ℕ) (e : Fin r ≃ PositionInside (zeroSet J gamma))
(f : (Fin r → O × N) → ℚ) :
average (fun sample : RawObservationFibre J name gamma observed =>
f (fun i => incidence name (sample.val.2 (e i).val))) =
average (fun fresh : Fin r → O × Fin 3 => f (fun i => incidence name (fresh i))) := by
calc
_ = average (fun fresh : InsideDraw (zeroSet J gamma) O =>
f (fun i => incidence name (fresh (e i)))) :=
raw_conditional_inside_average J name gamma observed hpos
(fun fresh => f (fun i => incidence name (fresh (e i))))
_ = _ := average_equiv (coordinateReindex e)
(fun fresh : Fin r → O × Fin 3 => f (fun i => incidence name (fresh i)))
theorem raw_conditional_event_reindex_real
{P D O N : Type} [Fintype P] [DecidableEq P]
[Fintype D] [Zero D] [DecidableEq D]
[Fintype O] [DecidableEq O] [DecidableEq N]
(J : Finset P) (name : O → Fin 3 → N) (gamma : RawCoefficients J D)
(observed : PositionOutside (zeroSet J gamma) → O × N)
(hpos : 0 < Fintype.card (RawObservationFibre J name gamma observed))
(r : ℕ) (e : Fin r ≃ PositionInside (zeroSet J gamma))
(event : (Fin r → O × N) → Prop) [DecidablePred event] :
((∑ sample : RawObservationFibre J name gamma observed,
if event (fun i => incidence name (sample.val.2 (e i).val)) then (1:ℝ) else 0) /
Fintype.card (RawObservationFibre J name gamma observed)) =
((∑ fresh : Fin r → O × Fin 3,
if event (fun i => incidence name (fresh i)) then (1:ℝ) else 0) /
Fintype.card (Fin r → O × Fin 3)) := by
have hQ := raw_conditional_incidence_reindex J name gamma observed hpos r e
(fun q => if event q then (1:ℚ) else 0)
have hR := congrArg (fun q : ℚ => (q : ℝ)) hQ
simpa only [average, Rat.cast_div, Rat.cast_sum, Rat.cast_natCast,
apply_ite, Rat.cast_one, Rat.cast_zero] using hR
end UniqueGamesTheorem.Soundness.ConditionalIncidences
end
end OAI
|