| import OAI.Combinatorics.BalancedRyser.Local.AlignedPointExposure |
| import OAI.Combinatorics.BalancedRyser.Local.LocalDirectionAssignments |
| |
| namespace OAI |
|
|
| namespace Balanced.Local.TokenExposure |
| |
| open AlignedListProbability TupleAlignment PartialAlignment |
| noncomputable section |
| attribute [local instance] Classical.propDecidable |
|
|
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
| variable {b κ : ℕ} |
|
|
| def tupleDirections (P : EqualityPattern b κ) |
| (e : DirectionToken b κ ↪ Direction F) (i : Fin b) : Fin 4 → Direction F := |
| fun j => e (Sum.inr (P.classOf i j)) |
| |
| omit [Field F] [Fintype F] [DecidableEq F] in |
| theorem tupleDirections_injective (P : EqualityPattern b κ) |
| (e : DirectionToken b κ ↪ Direction F) (i : Fin b) : |
| Function.Injective (tupleDirections P e i) := by |
| intro j k h |
| exact P.withinTuple i (Sum.inr.inj (e.injective h)) |
|
|
| omit [Field F] [Fintype F] [DecidableEq F] in |
| theorem tupleDirections_adjacent (P : EqualityPattern b κ) |
| (e : DirectionToken b κ ↪ Direction F) (i : Fin b) : |
| ∀ j, tupleDirections P e i j ≠ tupleDirections P e i (j+1) := |
| adjacent_ne_of_injective ⟨tupleDirections P e i, fun _ => false⟩ |
| (tupleDirections_injective P e i) |
|
|
| omit [Field F] [Fintype F] [DecidableEq F] in |
| theorem tupleDirections_ne_target (P : EqualityPattern b κ) |
| (e : DirectionToken b κ ↪ Direction F) (i : Fin b) : |
| ∀ j, tupleDirections P e i j ≠ e (Sum.inl i) := by |
| intro j h |
| have hbad := e.injective h |
| cases hbad |
|
|
| def pointKernel (P : EqualityPattern b κ) (bits : Bits b) (p : Point F) |
| (i : Fin b) (J : Finset (Fin 4)) (hJ : J.card ≤ 1) |
| (a : (DirectionToken b κ ↪ Direction F) → Fin 4 → Bool → F) |
| (e : DirectionToken b κ ↪ Direction F) : PMF (Fin 4 → Point F) := |
| generatedPointLaw (tupleDirections P e i) (tupleDirections_adjacent P e i) |
| (bits i) (e (Sum.inl i)) (tupleDirections_ne_target P e i) |
| (lineEval (e (Sum.inl i)) p) J hJ (a e) |
|
|
| theorem pointKernel_dependsOnExposed (P : EqualityPattern b κ) |
| (bits : Bits b) (p : Point F) (i : Fin b) |
| (J : Finset (Fin 4)) (hJ : J.card ≤ 1) |
| (K : Finset (DirectionToken b κ)) (htarget : Sum.inl i ∈ K) |
| (hold : ∀ j ∈ J, Sum.inr (P.classOf i j) ∈ K) |
| (a : (DirectionToken b κ ↪ Direction F) → Fin 4 → Bool → F) |
| (ha : ∀ e f, (∀ t ∈ K, e t = f t) → ∀ j ∈ J, a e j = a f j) : |
| InjectionExposure.DependsOnExposed K (pointKernel P bits p i J hJ a) := by |
| intro e f hef |
| have ht : e (Sum.inl i) = f (Sum.inl i) := hef _ htarget |
| have hd : ∀ j ∈ J, tupleDirections P e i j = tupleDirections P f i j := by |
| intro j hj |
| exact hef _ (hold j hj) |
| have hg : ∀ j, tupleDirections P f i j ≠ e (Sum.inl i) := by |
| simpa only [ht] using tupleDirections_ne_target P f i |
| have h := generatedPointLaw_agree_old |
| (tupleDirections P e i) (tupleDirections P f i) |
| (tupleDirections_adjacent P e i) (tupleDirections_adjacent P f i) |
| (bits i) (e (Sum.inl i)) (tupleDirections_ne_target P e i) hg |
| (lineEval (e (Sum.inl i)) p) J hJ (a e) (a f) hd (ha e f hef) |
| simpa only [pointKernel, ht] using h |
|
|
| theorem pointKernel_fixed_dependsOnExposed (P : EqualityPattern b κ) |
| (bits : Bits b) (p : Point F) (i : Fin b) |
| (J : Finset (Fin 4)) (hJ : J.card ≤ 1) |
| (K : Finset (DirectionToken b κ)) (htarget : Sum.inl i ∈ K) |
| (hold : ∀ j ∈ J, Sum.inr (P.classOf i j) ∈ K) |
| (a : Fin 4 → Bool → F) : |
| InjectionExposure.DependsOnExposed K (pointKernel P bits p i J hJ (fun _ => a)) := |
| pointKernel_dependsOnExposed P bits p i J hJ K htarget hold (fun _ => a) |
| (fun _ _ _ _ _ => rfl) |
|
|
| end |
| end Balanced.Local.TokenExposure |
| |
| end OAI |
|
|
