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e5a97c6 | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPreparedBind
namespace OAI
/-! # Adding the seven sequential generation stages without inflating exponents -/
noncomputable section
namespace LowSpaceSubsetSum.WordSeedMeasured
open WordSeedRealization
variable {w N M : ℕ} {α : Type} [Fintype α]
/-- Each completed stage releases scratch space and retains its bounded seed. -/
def Budgeted {S : SamplerCoupling α} {X : Realization w N S}
(E : Execution X) (B D k : ℕ) : Prop :=
E.Bounded (k*(B+2)) (k*(D+8)) (k*(D+8))
theorem budgeted_one {S : SamplerCoupling α} {X : Realization w N S}
(E : Execution X) (B D : ℕ) (h : E.Bounded B D D) : Budgeted E B D 1 := by
intro stream cursor
have hh := h stream cursor
exact ⟨by simpa only [Nat.one_mul] using hh.1.trans (Nat.le_add_right B 2),
by simpa only [Nat.one_mul] using hh.2.1.trans (Nat.le_add_right D 8),
by simpa only [Nat.one_mul] using hh.2.2.trans (Nat.le_add_right D 8)⟩
theorem budgeted_bind {S : SamplerCoupling α} {X : Realization w N S}
(E : Execution X) (A : α → Type) [∀ a, Fintype (A a)]
(T : ∀ a, SamplerCoupling (A a)) (Y : ∀ a, Realization w M (T a))
(F : ∀ a, Execution (Y a)) (B D k : ℕ) (hE : Budgeted E B D k)
(hF : ∀ a, (F a).Bounded B D D) :
Budgeted (bind E A T Y F) B D (k+1) := by
have h := bind_bounded E A T Y F _ _ _ _ _ _ hE hF
intro stream cursor
have hh := h stream cursor
refine ⟨hh.1.trans (by nlinarith),hh.2.1.trans ?_,hh.2.2.trans (by nlinarith)⟩
exact max_le (by nlinarith) (by nlinarith)
theorem budgeted_bindSeed {S : SamplerCoupling α} {X : Realization w N S}
(E : Execution X) (A : α → Type) [∀ a, Fintype (A a)]
(T : ∀ a, SamplerCoupling (A a)) (Y : ∀ a, Realization w M (T a))
(F : ∀ a : Accepted S, Execution (Y a.value)) (B D k : ℕ) (hE : Budgeted E B D k)
(hF : ∀ a, (F a).Bounded B D D) :
Budgeted (bindSeed E A T Y F) B D (k+1) := by
have h := bindSeed_bounded E A T Y F _ _ _ _ _ _ hE hF
intro stream cursor
have hh := h stream cursor
refine ⟨hh.1.trans (by nlinarith),hh.2.1.trans ?_,hh.2.2.trans (by nlinarith)⟩
exact max_le (by nlinarith) (by nlinarith)
end LowSpaceSubsetSum.WordSeedMeasured
end
end OAI
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