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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Storage.PermutationDigitsExecution
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.SignMaskExecution
namespace OAI
/-! # Materializing the bounded layout draw
An accepted permutation is stored as an indexed array, and the independent
sign bits are scanned into a bounded mask. Failed permutation draws halt
without using the coupling's ideal backup.
-/
noncomputable section
namespace LowSpaceSubsetSum.RepresentationLayoutExecution
variable {n : ℕ}
def run (R : RepresentationSetup n) (s : R.layoutCoupling.Seed) :
Option (Vector (Fin n) n × IndexedMask n) × ℕ :=
let digits := PermutationDigitsExecution.run n n s.1
match digits.value with
| none => (none, digits.reads + digits.tests + 1)
| some d =>
let v := PermutationArray.build n d
let signs := SignMaskExecution.run n s.2
(some (v.1, signs.1), digits.reads + digits.tests + v.2 + signs.2 + 3)
theorem value (R : RepresentationSetup n) (s : R.layoutCoupling.Seed)
(v : Vector (Fin n) n) (F : IndexedMask n) (h : (run R s).1 = some (v, F)) :
∃ e, R.layoutCoupling.bounded s = some (e, F.semantic) ∧
∀ i : Fin n, v[i.val] = e i := by
rcases s with ⟨first, second⟩
simp only [run] at h
cases hd : (PermutationDigitsExecution.run n n first).value with
| none => simp only [hd, reduceCtorEq] at h
| some d =>
simp only [hd, Option.some.injEq, Prod.mk.injEq] at h
rcases h with ⟨rfl, rfl⟩
refine ⟨PermutationArray.semantic n d, ?_, PermutationArray.build_value n d⟩
have hp := PermutationDigitsExecution.value n n first
rw [hd] at hp
have hp' : (permutationCoupling n n).bounded first = some (PermutationArray.semantic n d) := by
rw [← (SamplerExecution.permutation n n).value_eq]
exact hp.symm
have hi := (SamplerExecution.permutation n n).ideal_eq_of_some first
(PermutationArray.semantic n d) hp.symm
simp only [RepresentationSetup.layoutCoupling, SamplerCoupling.prod,
SamplerCoupling.map, SamplerCoupling.bind, hp', reduceCtorEq, ↓reduceIte,
signCoupling, bitArrayCoupling, Option.map_some]
rw [hi]
exact congrArg (fun mask => some (PermutationArray.semantic n d, mask))
(SignMaskExecution.value n second).symm
theorem complete (R : RepresentationSetup n) (s : R.layoutCoupling.Seed)
(e : Equiv.Perm (Fin n)) (F : Finset (Fin n))
(h : R.layoutCoupling.bounded s = some (e, F)) :
∃ v mask, (run R s).1 = some (v, mask) ∧ mask.semantic = F ∧
∀ i : Fin n, v[i.val] = e i := by
rcases s with ⟨first, second⟩
have hp := PermutationDigitsExecution.value n n first
cases hd : (PermutationDigitsExecution.run n n first).value with
| none =>
rw [hd] at hp
have hp' : (permutationCoupling n n).bounded first = none := by
rw [← (SamplerExecution.permutation n n).value_eq]
exact hp.symm
simp only [RepresentationSetup.layoutCoupling, SamplerCoupling.prod,
SamplerCoupling.map, SamplerCoupling.bind, hp', ↓reduceIte, Option.map_none,
reduceCtorEq] at h
| some d =>
rw [hd] at hp
have hp' : (permutationCoupling n n).bounded first = some (PermutationArray.semantic n d) := by
rw [← (SamplerExecution.permutation n n).value_eq]
exact hp.symm
have hi := (SamplerExecution.permutation n n).ideal_eq_of_some first
(PermutationArray.semantic n d) hp.symm
simp only [RepresentationSetup.layoutCoupling, SamplerCoupling.prod,
SamplerCoupling.map, SamplerCoupling.bind, hp', reduceCtorEq, ↓reduceIte,
signCoupling, bitArrayCoupling, Option.map_some] at h
have he : PermutationArray.semantic n d = e := hi.symm.trans (congrArg Prod.fst (Option.some.inj h))
have hF : bitsSubsetEquiv n second = F := congrArg Prod.snd (Option.some.inj h)
refine ⟨(PermutationArray.build n d).1, (SignMaskExecution.run n second).1, ?_, ?_, ?_⟩
· simp only [run, hd]
· exact (SignMaskExecution.value n second).trans hF
· rw [← he]
exact PermutationArray.build_value n d
theorem work (R : RepresentationSetup n) (s : R.layoutCoupling.Seed) :
(run R s).2 ≤ n * n ^ 3 + 10 * (n + 1) ^ 2 + 4 * n + 5 := by
have hd := PermutationDigitsExecution.counts n n s.1
simp only [run]
split
· dsimp only
omega
next d _ =>
have hv := PermutationArray.build_work n d
change _ + _ + _ + (4 * n + 2) + 3 ≤ _
omega
end LowSpaceSubsetSum.RepresentationLayoutExecution
end
end OAI
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