| import Mathlib |
| import OAI.Probability.BinarySweep.Trajectories.Trajectory |
| import OAI.Probability.BinarySweep.Conditional.PlacementBlocks |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open Irrep |
|
|
| variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
| (x : Placement H k 0) (g : ConditionalChoices H) |
|
|
| def augmentedPlacement (t : Fin (b+1)) : Placement H k t where |
| toFun i := ⟨gridPartialSweep bits g.val t (x i).val, by |
| rintro ⟨j,hj⟩ |
| have he := gridPartialSweep_path H g.val g.property t j |
| have hx := (gridPartialSweep bits g.val t).injective (he.trans hj) |
| exact (x i).property ⟨j,hx⟩⟩ |
| inj' i j hij := x.injective (Subtype.ext |
| ((gridPartialSweep bits g.val t).injective (congrArg Subtype.val hij))) |
|
|
| lemma augmentedPosition_left (t : Fin (b+1)) (i : Fin h) : |
| (augmentByChoice H x g).position t (Fin.castAdd k i) = H.position t i := by |
| change gridPartialSweep bits g.val t (Fin.append _ _ (Fin.castAdd k i)) = _ |
| rw [Fin.append_left,gridPartialSweep_path H g.val g.property] |
|
|
| lemma augmentedPosition_right (t : Fin (b+1)) (i : Fin k) : |
| (augmentByChoice H x g).position t (Fin.natAdd h i) = (augmentedPlacement H x g t i).val := by |
| change gridPartialSweep bits g.val t (Fin.append _ _ (Fin.natAdd h i)) = _ |
| rw [Fin.append_right]; rfl |
| |
| lemma augmentedPlacement_zero : augmentedPlacement H x g 0 = x := by |
| apply Function.Embedding.ext |
| intro index |
| apply Subtype.ext |
| change gridPartialSweep bits g.val 0 (x index).val = (x index).val |
| rw [gridPartialSweep_zero, Equiv.Perm.one_apply] |
|
|
| lemma augmentedPlacement_last : augmentedPlacement H x g (Fin.last b) = placementBijection H k g x := by |
| apply Function.Embedding.ext |
| intro index |
| apply Subtype.ext |
| change gridPartialSweep bits g.val (Fin.last b) (x index).val = |
| gridSweep bits g.val (x index).val |
| rw [gridPartialSweep_last] |
|
|
| def augmentationFreeEquiv (t : Fin (b+1)) : |
| FreeSlot (augmentByChoice H x g) t ≃ PlacementComplement (augmentedPlacement H x g t) where |
| toFun w := ⟨⟨w.val, by |
| rintro ⟨i,hi⟩ |
| exact w.property ⟨Fin.castAdd k i,(augmentedPosition_left H x g t i).trans hi⟩⟩,by |
| rintro ⟨i,hi⟩ |
| exact w.property ⟨Fin.natAdd h i,(augmentedPosition_right H x g t i).trans |
| (congrArg Subtype.val hi)⟩⟩ |
| invFun w := ⟨w.val.val,by |
| rintro ⟨i,hi⟩ |
| induction i using Fin.addCases with |
| | left i => |
| rw [augmentedPosition_left] at hi |
| exact w.val.property ⟨i,hi⟩ |
| | right i => |
| rw [augmentedPosition_right] at hi |
| exact w.property ⟨i,Subtype.ext hi⟩⟩ |
| left_inv w := rfl |
| right_inv w := rfl |
|
|
| lemma augmentationFreeEquiv_val (t : Fin (b+1)) (w : FreeSlot (augmentByChoice H x g) t) : |
| (augmentationFreeEquiv H x g t w).val.val = w.val := rfl |
|
|
| lemma augmented_gridSweep (g' : ConditionalChoices (augmentByChoice H x g)) |
| (w : FreeSlot (augmentByChoice H x g) 0) : |
| (augmentationFreeEquiv H x g (Fin.last b) |
| (remainingBijection (augmentByChoice H x g) g'.val g'.property w)).val = |
| remainingBijection H g'.val |
| ((augmentByChoice_extension H x g).pathEvent_mono g'.property) |
| (augmentationFreeEquiv H x g 0 w).val := rfl |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| end OAI |
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|
