openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Search /BudgetedPrefixCompletion.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedPrefixAllowance | |
| namespace OAI | |
| namespace LowSpaceSubsetSum.BudgetedPrefixController | |
| variable {n : ℕ} {I : Instance n} {C : ℕ} | |
| /-- Paying the full source-prefix ledger guarantees that early-YES execution | |
| finishes; the allowance does not have to be computed at runtime. -/ | |
| theorem run_complete | |
| (collect : StoredRestriction n → ℕ → Option (StoredEntry.Dictionary I C × ℕ) × ℕ) | |
| (pureCollect : StoredRestriction n → StoredEntry.Dictionary I C) | |
| (check : Array (StoredEntry n) → ℕ → Option (Bool × ℕ) × ℕ) | |
| (collectBill : StoredRestriction n → ℕ) (checkBill : Array (StoredEntry n) → ℕ) | |
| (hc : ∀ P b, collectBill P ≤ b → ∃ left, (collect P b).1 = some (pureCollect P, left)) | |
| (hcr : ∀ P b d left, (collect P b).1 = some (d, left) → d = pureCollect P ∧ b ≤ left + collectBill P) | |
| (hk : ∀ a b, checkBill a ≤ b → ∃ v left, (check a b).1 = some (v, left)) | |
| (hkr : ∀ a b v left, (check a b).1 = some (v, left) → b ≤ left + checkBill a) | |
| (order : List (Fin n)) (P : StoredRestriction n) (budget : ℕ) | |
| (hb : allowance pureCollect collectBill checkBill order P ≤ budget) : | |
| ∃ v left, (run collect check order P budget).1 = some (v, left) := by | |
| induction order generalizing P budget with | |
| | nil => | |
| have hhead : StoredPrefixController.header n ≤ budget := by | |
| unfold allowance at hb | |
| split at hb <;> omega | |
| have hbill : collectBill P ≤ budget - StoredPrefixController.header n := by | |
| unfold allowance at hb | |
| split at hb <;> omega | |
| obtain ⟨left, hcollect⟩ := hc P (budget - StoredPrefixController.header n) hbill | |
| have hr := (hcr P _ _ _ hcollect).2 | |
| by_cases ho : (pureCollect P).val.keys.overflow = true | |
| · exact ⟨false, left, by rw [run, ite_eq_left hhead]; simp only [↓reduceIte, hcollect, ho]⟩ | |
| · have hcheck : checkBill (pureCollect P).val.records ≤ left := by | |
| simp only [allowance, ho, Bool.false_eq_true, ↓reduceIte] at hb | |
| omega | |
| obtain ⟨v, rest, hchecked⟩ := hk (pureCollect P).val.records left hcheck | |
| exact ⟨v, rest, by rw [run, ite_eq_left hhead]; simp only [↓reduceIte, hcollect, ho, Bool.false_eq_true, hchecked]⟩ | |
| | cons i order ih => | |
| have hhead : StoredPrefixController.header n ≤ budget := by | |
| unfold allowance at hb | |
| split at hb <;> omega | |
| have hbill : collectBill P ≤ budget - StoredPrefixController.header n := by | |
| unfold allowance at hb | |
| split at hb <;> omega | |
| obtain ⟨left, hcollect⟩ := hc P (budget - StoredPrefixController.header n) hbill | |
| have hr := (hcr P _ _ _ hcollect).2 | |
| by_cases ho : (pureCollect P).val.keys.overflow = true | |
| · have hsum : allowance pureCollect collectBill checkBill order (P.force i false).1 + | |
| allowance pureCollect collectBill checkBill order (P.force i true).1 ≤ left := by | |
| simp only [allowance, ho, ↓reduceIte] at hb | |
| omega | |
| obtain ⟨v, rest, hl⟩ := ih (P.force i false).1 left (by omega) | |
| have hleft := run_remaining collect pureCollect check collectBill checkBill hcr hkr | |
| order (P.force i false).1 left v rest hl | |
| cases v | |
| · obtain ⟨v', finalBudget, hright⟩ := ih (P.force i true).1 rest (by omega) | |
| exact ⟨v', finalBudget, by rw [run, ite_eq_left hhead]; simp only [↓reduceIte, hcollect, ho, hl, hright]⟩ | |
| · exact ⟨true, rest, by rw [run, ite_eq_left hhead]; simp only [↓reduceIte, hcollect, ho, hl]⟩ | |
| · have hcheck : checkBill (pureCollect P).val.records ≤ left := by | |
| simp only [allowance, ho, Bool.false_eq_true, ↓reduceIte] at hb | |
| omega | |
| obtain ⟨v, rest, hchecked⟩ := hk (pureCollect P).val.records left hcheck | |
| exact ⟨v, rest, by rw [run, ite_eq_left hhead]; simp only [↓reduceIte, hcollect, ho, Bool.false_eq_true, hchecked]⟩ | |
| end LowSpaceSubsetSum.BudgetedPrefixController | |
| end OAI | |