openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordSeedAdditiveBudget.lean
| 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 | |