| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedPrefixController |
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.BudgetedSeparatorFlow |
|
|
| namespace OAI |
|
|
| namespace LowSpaceSubsetSum.BudgetedPrefixController |
|
|
| open BudgetedSeparatorFlow (Account) |
| variable {n : ℕ} {I : Instance n} {C : ℕ} |
|
|
| private theorem force_bound (P : StoredRestriction n) (i : Fin n) : |
| (P.force i false).2 + (P.force i true).2 + 8 ≤ StoredPrefixController.header n := by |
| have h₀ := P.force_work i false |
| have h₁ := P.force_work i true |
| unfold StoredPrefixController.header |
| omega |
|
|
| theorem run_account |
| (collect : StoredRestriction n → ℕ → Option (StoredEntry.Dictionary I C × ℕ) × ℕ) |
| (check : Array (StoredEntry n) → ℕ → Option (Bool × ℕ) × ℕ) |
| (hc : ∀ P b, Account b (collect P b)) (hk : ∀ a b, Account b (check a b)) |
| (order : List (Fin n)) (P : StoredRestriction n) (budget : ℕ) : |
| Account budget (run collect check order P budget) := by |
| have hh : 8 ≤ StoredPrefixController.header n := by unfold StoredPrefixController.header; omega |
| induction order generalizing P budget with |
| | nil => |
| rw [run] |
| split |
| next hb => |
| generalize htdef : collect P (budget - StoredPrefixController.header n) = task |
| have htask := hc P (budget - StoredPrefixController.header n) |
| rw [htdef] at htask |
| cases ht : task.1 with |
| | none => |
| simp only [ht, BudgetedSeparatorFlow.Account] |
| constructor |
| · exact le_trans (Nat.add_le_add_right htask.1 2) (by omega) |
| · intro v left h |
| contradiction |
| | some p => |
| rcases p with ⟨d, left⟩ |
| have hts := htask.2 d left ht |
| simp only [ht, BudgetedSeparatorFlow.Account] |
| split |
| · constructor |
| · dsimp only; omega |
| · intro v rest h |
| cases h |
| dsimp only |
| omega |
| · have hchecked := hk d.val.records left |
| have hw := hchecked.1 |
| constructor |
| · change task.2 + (check d.val.records left).2 + 4 ≤ _ |
| omega |
| · intro v rest h |
| have hs := hchecked.2 v rest h |
| change rest ≤ budget ∧ task.2 + (check d.val.records left).2 + 4 ≤ budget - rest |
| omega |
| next _ => |
| constructor |
| · dsimp only; omega |
| · intro v left h |
| contradiction |
| | cons i order ih => |
| rw [run] |
| split |
| next hb => |
| generalize htdef : collect P (budget - StoredPrefixController.header n) = task |
| have htask := hc P (budget - StoredPrefixController.header n) |
| rw [htdef] at htask |
| cases ht : task.1 with |
| | none => |
| simp only [ht, BudgetedSeparatorFlow.Account] |
| constructor |
| · exact le_trans (Nat.add_le_add_right htask.1 2) (by omega) |
| · intro v left h |
| contradiction |
| | some p => |
| rcases p with ⟨d, available⟩ |
| have hts := htask.2 d available ht |
| simp only [ht, BudgetedSeparatorFlow.Account] |
| split |
| · have hf := force_bound P i |
| generalize hldef : run collect check order (P.force i false).1 available = left |
| have hl := ih (P.force i false).1 available |
| rw [hldef] at hl |
| have hlw := hl.1 |
| cases he : left.1 with |
| | none => |
| constructor |
| · change task.2 + (P.force i false).2 + left.2 + 6 ≤ _ |
| omega |
| · intro v rest h |
| contradiction |
| | some p => |
| rcases p with ⟨v, rest⟩ |
| have hls := hl.2 v rest he |
| cases v |
| · dsimp only |
| generalize hrdef : run collect check order (P.force i true).1 rest = right |
| have hr := ih (P.force i true).1 rest |
| rw [hrdef] at hr |
| have hrw := hr.1 |
| constructor |
| · omega |
| · intro v remaining h |
| have hrs := hr.2 v remaining h |
| change remaining ≤ budget ∧ |
| task.2 + (P.force i false).2 + left.2 + (P.force i true).2 + right.2 + 8 ≤ budget - remaining |
| omega |
| · constructor |
| · change task.2 + (P.force i false).2 + left.2 + 6 ≤ _ |
| omega |
| · intro v remaining h |
| cases h |
| change rest ≤ budget ∧ task.2 + (P.force i false).2 + left.2 + 6 ≤ budget - rest |
| omega |
| · have hchecked := hk d.val.records available |
| have hw := hchecked.1 |
| constructor |
| · change task.2 + (check d.val.records available).2 + 4 ≤ _ |
| omega |
| · intro v rest h |
| have hs := hchecked.2 v rest h |
| change rest ≤ budget ∧ task.2 + (check d.val.records available).2 + 4 ≤ budget - rest |
| omega |
| next _ => |
| constructor |
| · dsimp only; omega |
| · intro v left h |
| contradiction |
|
|
| end LowSpaceSubsetSum.BudgetedPrefixController |
|
|
| end OAI |
|
|
