| import OAI.Combinatorics.BalancedRyser.Local.AlignedGlobalHistory |
| import OAI.Combinatorics.BalancedRyser.Local.AlignedUsedObservable |
| import OAI.Combinatorics.BalancedRyser.Local.InjectionAuxiliaryExposure |
| import OAI.Combinatorics.BalancedRyser.Local.LocalDirectionAssignments |
|
|
| namespace OAI |
|
|
| namespace Balanced.Local.AlignedGlobal |
|
|
| open RootChildPartition InjectionExposure |
| noncomputable section |
| open scoped Classical |
|
|
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
| variable {b κ : ℕ} |
|
|
| abbrev PrefixData (F : Type*) {b κ : ℕ} (P : EqualityPattern b κ) |
| (S : Finset (Fin b)) := |
| ({i : Fin b // i ∈ S} → Fin 4 → Point F) × |
| ({c : Fin (4*b-κ) // c ∈ S.biUnion P.classSet} → Bool → Line F) |
|
|
| def prefixObservation (P : EqualityPattern b κ) (S : Finset (Fin b)) |
| (a : DirectionAssignment F b κ) (bits : Bits b) (M : MergedPairs F) : |
| PrefixData F P S := |
| ((fun i j => prescribedPoint P a bits M (i.1,j)), |
| fun c s => constituent M (a.2 c.1) s) |
|
|
| theorem generated_prefixObservation_eq (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (R S : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) |
| (hS : S ⊆ R ∨ R ⊆ S) |
| (a a' : DirectionAssignment F b κ) (ha : Eligible P a) (ha' : Eligible P a') |
| (bits : Bits b) (p : Point F) (x : FreeIndex P R → F) |
| (ht : ∀ i ∈ S, a.1 i = a'.1 i) |
| (hd : ∀ c ∈ S.biUnion P.classSet, a.2 c = a'.2 c) : |
| prefixObservation P S a bits (generatedZero P hκ R hD a ha bits p x) = |
| prefixObservation P S a' bits (generatedZero P hκ R hD a' ha' bits p x) := by |
| have h := generated_history_agree P hκ R S hD hS a a' ha ha' bits p x |
| (fun _ _ => 0) (fun _ _ => 0) ht hd |
| apply Prod.ext |
| · funext i j |
| exact h.1 i.1 i.2 j |
| · funext c s |
| apply Prod.ext |
| · exact hd c.1 c.2 |
| · exact congrFun (h.2 c.1 c.2) s |
|
|
| def prefixNoise (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (R S : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) |
| (bits : Bits b) (p : Point F) (e : DirectionToken b κ ↪ Direction F) : |
| PMF ((FreeIndex P R → F) × PrefixData F P S) := |
| let a := tokenInjectionEquiv F b κ e |
| (PMF.uniformOfFintype (FreeIndex P R → F)).map fun x => |
| (x, prefixObservation P S a.1 bits |
| (generatedZero P hκ R hD a.1 (allTokensDistinct_eligible P a.1 a.2) bits p x)) |
|
|
| theorem prefixNoise_dependsOnExposed (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (R S : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) |
| (hS : S ⊆ R ∨ R ⊆ S) (bits : Bits b) (p : Point F) |
| (K : Finset (DirectionToken b κ)) |
| (hKt : ∀ i ∈ S, Sum.inl i ∈ K) |
| (hKg : ∀ c ∈ S.biUnion P.classSet, Sum.inr c ∈ K) : |
| DependsOnExposed K (prefixNoise P hκ R S hD bits p) := by |
| intro e f hef |
| dsimp only [prefixNoise] |
| congr 1 |
| funext x |
| apply Prod.ext |
| · rfl |
| apply generated_prefixObservation_eq P hκ R S hD hS |
| · intro i hi |
| exact hef _ (hKt i hi) |
| · intro c hc |
| exact hef _ (hKg c hc) |
|
|
| def currentNoise (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (R S : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) |
| (bits : Bits b) (p : Point F) (i : Fin b) |
| (e : DirectionToken b κ ↪ Direction F) : |
| PMF ((FreeIndex P R → F) × (PrefixData F P S × (Fin 4 → Point F))) := |
| let a := tokenInjectionEquiv F b κ e |
| (PMF.uniformOfFintype (FreeIndex P R → F)).map fun x => |
| let M := generatedZero P hκ R hD a.1 (allTokensDistinct_eligible P a.1 a.2) bits p x |
| (x, prefixObservation P S a.1 bits M, fun j => prescribedPoint P a.1 bits M (i,j)) |
|
|
| theorem currentNoise_dependsOnExposed (P : EqualityPattern b κ) (hκ : κ ≤ 2) |
| (R S : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ) |
| (hS : S ⊆ R ∨ R ⊆ S) (bits : Bits b) (p : Point F) (i : Fin b) |
| (hphase : i ∈ R ∨ R ⊆ S) |
| (K : Finset (DirectionToken b κ)) |
| (hKt : ∀ r ∈ S, Sum.inl r ∈ K) |
| (hKg : ∀ c ∈ S.biUnion P.classSet, Sum.inr c ∈ K) |
| (hi : Sum.inl i ∈ K) : |
| DependsOnExposed K (currentNoise P hκ R S hD bits p i) := by |
| intro e f hef |
| let a := tokenInjectionEquiv F b κ e |
| let a' := tokenInjectionEquiv F b κ f |
| let ha := allTokensDistinct_eligible P a.1 a.2 |
| let ha' := allTokensDistinct_eligible P a'.1 a'.2 |
| have ht : ∀ r ∈ S, a.1.1 r = a'.1.1 r := fun r hr => hef _ (hKt r hr) |
| have hd : ∀ c ∈ S.biUnion P.classSet, a.1.2 c = a'.1.2 c := |
| fun c hc => hef _ (hKg c hc) |
| have hti : a.1.1 i = a'.1.1 i := hef _ hi |
| dsimp only [currentNoise] |
| congr 1 |
| funext x |
| apply Prod.ext |
| · rfl |
| apply Prod.ext |
| · exact generated_prefixObservation_eq P hκ R S hD hS a.1 a'.1 ha ha' |
| bits p x ht hd |
| · funext j |
| rcases hphase with hir | hRS |
| · apply free_point_eq_of_coordinate_eq P R a.1 a'.1 bits p |
| ((coordinateEquiv P hκ R hD a.1 ha bits p).symm (x,fun _ _ => 0)) |
| ((coordinateEquiv P hκ R hD a'.1 ha' bits p).symm (x,fun _ _ => 0)) |
| (inverse_coordinates_fst_eq P hκ R hD a.1 a'.1 ha ha' bits p x |
| (fun _ _ => 0) (fun _ _ => 0)) ⟨(i,j), ?_⟩ hti |
| simp only [mem_dependentOccurrences, not_and] |
| exact fun hn => (hn hir).elim |
| · apply generated_point_eq_of_root_history P hκ R hD a.1 a'.1 ha ha' |
| bits p x (fun _ _ => 0) (fun _ _ => 0) |
| (fun r hr => ht r (hRS hr)) ?_ (i,j) hti |
| intro c hc |
| apply hd c |
| obtain ⟨r, hr, hcr⟩ := Finset.mem_biUnion.mp hc |
| exact Finset.mem_biUnion.mpr ⟨r, hRS hr, hcr⟩ |
|
|
| end |
| end Balanced.Local.AlignedGlobal |
|
|
| end OAI |
|
|
