openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Probability /FiniteBufferRanges.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryGatheredValues | |
| namespace OAI | |
| /-! The default-zero finite buffers preserve the nonnegative bounds of | |
| paired initial/history coordinates when the initial buffer covers them. -/ | |
| namespace EditApproximation | |
| theorem finiteAnswerTable_eq_zero_of_not_mem {κ : Type*} [DecidableEq κ] | |
| (keys : List κ) (values : List ℚ) (query : κ) (hq : query ∉ keys) : | |
| finiteAnswerTable keys values query = 0 := by | |
| induction keys generalizing values with | |
| | nil => rfl | |
| | cons key keys ih => | |
| cases values with | |
| | nil => rfl | |
| | cons value values => | |
| have hk : query ≠ key := fun he => hq (List.mem_cons.mpr (Or.inl he)) | |
| have ht : query ∉ keys := fun he => hq (List.mem_cons_of_mem key he) | |
| simpa only [finiteAnswerTable, ite_eq_right hk] using ih values ht | |
| theorem finiteAnswerTable_nonneg {κ : Type*} [DecidableEq κ] | |
| (keys : List κ) (values : List ℚ) (query : κ) (hv : ∀ value ∈ values, 0 ≤ value) : | |
| 0 ≤ finiteAnswerTable keys values query := by | |
| induction keys generalizing values with | |
| | nil => exact le_rfl | |
| | cons key keys ih => | |
| cases values with | |
| | nil => exact le_rfl | |
| | cons value values => | |
| unfold finiteAnswerTable | |
| split_ifs | |
| · exact hv value List.mem_cons_self | |
| · exact ih values (fun v h => hv v (List.mem_cons_of_mem _ h)) | |
| theorem finiteAnswerTable_map_le_of_subset {κ : Type*} [DecidableEq κ] | |
| (keys initialKeys : List κ) (history initial : κ → ℚ) | |
| (hkeys : ∀ key ∈ keys, key ∈ initialKeys) | |
| (hi : ∀ key, 0 ≤ initial key) | |
| (hle : ∀ key, history key ≤ initial key) (query : κ) : | |
| finiteAnswerTable keys (keys.map history) query ≤ | |
| finiteAnswerTable initialKeys (initialKeys.map initial) query := by | |
| by_cases hq : query ∈ keys | |
| · rw [finiteAnswerTable_map keys history query hq, | |
| finiteAnswerTable_map initialKeys initial query (hkeys query hq)] | |
| exact hle query | |
| · rw [finiteAnswerTable_eq_zero_of_not_mem keys _ query hq] | |
| apply finiteAnswerTable_nonneg | |
| intro value hv | |
| obtain ⟨key, _, rfl⟩ := List.mem_map.mp hv | |
| exact hi key | |
| end EditApproximation | |
| end OAI | |