File size: 3,290 Bytes
9b6e014 | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | 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
|