openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordSeedPermutationExecution.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedRangeInitialization | |
| namespace OAI | |
| /-! # Measured permutation seed generation | |
| Each range call computes its modulus, draws and retains the actual candidate | |
| tuple, and caches its accepted digit. Permutation values are composition | |
| closures; their application or array materialization remains a separate | |
| measured operation, as in `PermutationArray`. | |
| -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.WordSeedMeasured | |
| def permutation (w n : ℕ) : (k : ℕ) → (hw : Nat.clog 2 k ≤ w) → | |
| Execution (WordSeedRealization.permutation w n k hw) | |
| | 0, _ => by | |
| exact reprogram (pure (w := w) (Equiv.refl (Fin 0))) (by | |
| unfold WordSeedRealization.permutation | |
| exact mpr_program (Nat.zero_mul (n^3)) _) | |
| | k+1, hw => by | |
| have hk : Nat.clog 2 k ≤ w := | |
| (Nat.clog_mono_right 2 (by omega : k ≤ k+1)).trans hw | |
| exact reprogram | |
| (map (prod (fin w n (k+1) hw) (permutation w n k hk)) | |
| Equiv.Perm.decomposeFin.symm | |
| (fun p => ((Equiv.Perm.decomposeFin.symm p,6),8)) (fun _ => rfl)) (by | |
| conv_lhs => rw [WordSeedRealization.permutation] | |
| exact mpr_program (w := w) (α := Equiv.Perm (Fin (k+1))) | |
| (by simp only [Nat.add_mul,Nat.one_mul,Nat.add_comm] : | |
| (k+1)*n^3 = n^3+k*n^3) _) | |
| theorem permutation_bounded (w n : ℕ) : ∀ (k : ℕ) (hw : Nat.clog 2 k ≤ w), | |
| (permutation w n k hw).Bounded | |
| (k*(13*n^3+5*k+64)+1) ((k+1)*(2*n^3+256)) ((k+1)*(n^3+64)) := by | |
| intro k | |
| induction k with | |
| | zero => | |
| intro hw stream cursor | |
| change 1 ≤ _ ∧ 4 ≤ _ ∧ 4 ≤ _ | |
| constructor | |
| · omega | |
| · constructor <;> omega | |
| | succ k ih => | |
| intro hw | |
| have hk : Nat.clog 2 k ≤ w := | |
| (Nat.clog_mono_right 2 (by omega : k ≤ k+1)).trans hw | |
| have ht := ih hk | |
| have hf := fin_bounded w n (k+1) hw | |
| have hp := prod_bounded (fin w n (k+1) hw) (permutation w n k hk) | |
| _ _ _ _ _ _ hf ht | |
| have hm := map_bounded _ Equiv.Perm.decomposeFin.symm | |
| (fun p => ((Equiv.Perm.decomposeFin.symm p,6),8)) (fun _ => rfl) | |
| _ _ _ 6 8 hp (fun _ => le_rfl) (fun _ => le_rfl) | |
| have hclog : Nat.clog 2 (k+1) ≤ k+1 := | |
| Nat.clog_le_of_le_pow Nat.lt_two_pow_self.le | |
| intro stream cursor | |
| have h := hm stream cursor | |
| change _ ≤ (k+1)*(13*n^3+5*(k+1)+64)+1 ∧ | |
| _ ≤ (k+1+1)*(2*n^3+256) ∧ _ ≤ (k+1+1)*(n^3+64) | |
| refine ⟨h.1.trans (by nlinarith),h.2.1.trans ?_,h.2.2.trans (by nlinarith)⟩ | |
| apply max_le | |
| · apply max_le | |
| · apply max_le <;> nlinarith | |
| · nlinarith | |
| · nlinarith | |
| end LowSpaceSubsetSum.WordSeedMeasured | |
| end | |
| end OAI | |