openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordSeedMeasuredRejection.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedMeasured | |
| namespace OAI | |
| /-! # Measured rejection as a cached seed-generator primitive -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.WordSeedMeasured | |
| open WordSeedRealization | |
| variable {w k : ℕ} (A : Fin (2^k) → Prop) [DecidablePred A] | |
| [Nonempty {a : Fin (2^k) // A a}] | |
| theorem rejectionSeed_backup (N : ℕ) (f : Fin N → Fin (2^k)) | |
| (s : SuccessfulSeed (firstAcceptCoupling A N)) (h : rejectionSeed A N f = some s) : | |
| (firstAcceptCoupling A N).bounded s.val = some s.val.2 := by | |
| unfold rejectionSeed at h | |
| split at h | |
| · contradiction | |
| next a ha => | |
| cases h | |
| exact ha | |
| def cacheRejection (N : ℕ) (f : Fin N → Fin (2^k)) | |
| (out : Option (SuccessfulSeed (firstAcceptCoupling A N))) | |
| (ho : out = rejectionSeed A N f) : Option (Accepted (firstAcceptCoupling A N)) := | |
| out.pmap (fun s hs => | |
| ⟨s,s.val.2,Option.some.inj ((rejectionSeed_backup A N f s (ho.symm.trans hs)).symm.trans | |
| (successful_agrees s))⟩) (fun _ h => h) | |
| theorem cacheRejection_seed (N : ℕ) (f : Fin N → Fin (2^k)) | |
| (out : Option (SuccessfulSeed (firstAcceptCoupling A N))) | |
| (ho : out = rejectionSeed A N f) : | |
| (cacheRejection A N f out ho).map Accepted.seed = out := by | |
| cases out <;> rfl | |
| variable (evaluate : Fin (2^k) → (Bool × ℕ) × ℕ) | |
| (he : ∀ a, (evaluate a).1.1 = true ↔ A a) | |
| def rejectionRun (stream : ℕ → Fin (2^w)) (modulus : ℕ) (hm : modulus = 2^k) | |
| (N cursor : ℕ) : Result (firstAcceptCoupling A N) := | |
| let r := WordSeedRejectionExecution.run A evaluate he stream modulus hm N cursor | |
| ⟨cacheRejection A N (fun i => ExactWordDraw.low w k (stream (cursor+i.val))) | |
| r.seed (WordSeedRejectionExecution.run_seed A evaluate he stream modulus hm N cursor), | |
| r.cursor,r.work+2,r.cells+8,N+8⟩ | |
| def rejection (hk : k ≤ w) (modulus : ℕ) (hm : modulus = 2^k) (N : ℕ) : | |
| Execution (WordSeedRealization.rejection w k hk N A) where | |
| run stream cursor := rejectionRun A evaluate he stream modulus hm N cursor | |
| value stream cursor := by | |
| dsimp only [rejectionRun] | |
| rw [cacheRejection_seed] | |
| exact WordSeedRejectionExecution.program_value A evaluate he hk stream modulus hm N cursor | |
| theorem rejection_bounded (hk : k ≤ w) (modulus : ℕ) (hm : modulus = 2^k) | |
| (N K H : ℕ) (hK : ∀ a, (evaluate a).1.2 ≤ K) (hH : ∀ a, (evaluate a).2 ≤ H) : | |
| (rejection A evaluate he hk modulus hm N).Bounded | |
| (N*(K+11)+11) (2*N+H+64) (N+8) := by | |
| intro stream cursor | |
| have hw := WordSeedRejectionExecution.run_work A evaluate he stream modulus hm K N cursor hK | |
| have hc := WordSeedRejectionExecution.run_cells A evaluate he stream modulus hm H N cursor hH | |
| change _+2 ≤ _ ∧ _+8 ≤ _ ∧ N+8 ≤ N+8 | |
| exact ⟨by omega,by omega,le_rfl⟩ | |
| end LowSpaceSubsetSum.WordSeedMeasured | |
| end | |
| end OAI | |