File size: 4,805 Bytes
5d5048e | 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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationSurvival
import OAI.Combinatorics.LowSpaceSubsetSum.Counting.SamplerExecution
namespace OAI
/-! # Canonical consecutive labels for representation partitions
The base labeling is determined by the sorted mixer domain. Randomly
permuting these labels gives the analyzed uniform balanced partition, with
no choice of an arbitrary initial coloring required by the runtime.
-/
noncomputable section
namespace LowSpaceSubsetSum.CanonicalPartition
private def rankFiber (m g : ℕ) (j : Fin m) :
{x : Fin (m * g) // (finProdFinEquiv.symm x).1 = j} ≃ Fin g where
toFun x := (finProdFinEquiv.symm x.1).2
invFun k := ⟨finProdFinEquiv (j, k), by simp⟩
left_inv x := by
apply Subtype.ext
apply finProdFinEquiv.symm.injective
change finProdFinEquiv.symm (finProdFinEquiv (j, (finProdFinEquiv.symm x.val).2)) =
finProdFinEquiv.symm x.val
rw [Equiv.symm_apply_apply]
exact Prod.ext x.property.symm rfl
right_inv k := by simp
variable {n : ℕ}
theorem size_eq (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
(R.orderedMixers z i).card = RepresentationSetup.blockCount i * (2 * RepresentationSetup.blockHalf n i) := by
rw [R.orderedMixers_card, ← R.blockCount_half i]
ring
def ranks (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
↥(R.orderedMixers z i) ≃ Fin (RepresentationSetup.blockCount i * (2 * RepresentationSetup.blockHalf n i)) :=
((R.orderedMixers z i).orderIsoOfFin (size_eq R z i)).symm.toEquiv
def labels (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3)
(a : ↥(R.orderedMixers z i)) : Fin (RepresentationSetup.blockCount i) :=
(finProdFinEquiv.symm (ranks R z i a)).1
private def labelFiber (R : RepresentationSetup n) (z : RepresentationSetup.Layout n)
(i : Fin 3) (j : Fin (RepresentationSetup.blockCount i)) :
{a : ↥(R.orderedMixers z i) // labels R z i a = j} ≃
Fin (2 * RepresentationSetup.blockHalf n i) :=
((ranks R z i).subtypeEquivOfSubtype
(p := fun x => (finProdFinEquiv.symm x).1 = j)).trans (rankFiber _ _ j)
def base (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
R.Partition z i :=
⟨labels R z i, by
funext j
exact (Fintype.card_congr (labelFiber R z i j)).trans (Fintype.card_fin _)⟩
def coupling (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
SamplerCoupling (R.Partition z i) :=
((permutationCoupling n (RepresentationSetup.blockCount i * (2 * RepresentationSetup.blockHalf n i))).map
(Equiv.permCongr (ranks R z i).symm)).map (coloringOfPermutation (base R z i))
theorem coupling_uniform (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
(coupling R z i).idealLaw = @FiniteLaw.uniform _ inferInstance ⟨base R z i⟩ := by
rw [coupling, SamplerCoupling.map_idealLaw, SamplerCoupling.map_idealLaw,
permutationCoupling_uniform, FiniteLaw.map_equiv_uniform]
exact coloringOfPermutation_uniform (base R z i)
theorem coupling_failure (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
(coupling R z i).failure ≤ mixerSize n / (2 : ℝ) ^ (n ^ 3) := by
rw [coupling, SamplerCoupling.map_failure, SamplerCoupling.map_failure]
have h := permutationCoupling_failure n
(RepresentationSetup.blockCount i * (2 * RepresentationSetup.blockHalf n i))
have hc : RepresentationSetup.blockCount i * (2 * RepresentationSetup.blockHalf n i) = mixerSize n := by
rw [← size_eq R z i, R.orderedMixers_card, RepresentationSetup.halfSize]
unfold mixerSize
omega
simpa only [hc] using h
def execution (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
SamplerExecution.Execution (coupling R z i) :=
SamplerExecution.map
(SamplerExecution.map
(SamplerExecution.permutation n (RepresentationSetup.blockCount i *
(2 * RepresentationSetup.blockHalf n i)))
(Equiv.permCongr (ranks R z i).symm))
(coloringOfPermutation (base R z i))
theorem execution_bounded (R : RepresentationSetup n) (z : RepresentationSetup.Layout n) (i : Fin 3) :
(execution R z i).Bounded (mixerSize n * n ^ 3) 0 := by
have hc : RepresentationSetup.blockCount i * (2 * RepresentationSetup.blockHalf n i) = mixerSize n := by
rw [← size_eq R z i, R.orderedMixers_card, RepresentationSetup.halfSize]
unfold mixerSize
omega
unfold execution
apply SamplerExecution.map_bounded
apply SamplerExecution.map_bounded
simpa only [hc] using SamplerExecution.permutation_bounded n
(RepresentationSetup.blockCount i * (2 * RepresentationSetup.blockHalf n i))
end LowSpaceSubsetSum.CanonicalPartition
end
end OAI
|