| import AUXLib.Arithmetic |
| import SimpleC.SL.SeparationLogic |
| import ListLib.General.Length |
|
|
| namespace Data_structures.binary_indexed_tree.lean |
|
|
| open AUXLib |
|
|
| def FenwickLowbit (x : Int) : Int := Z.land x (-x) |
|
|
| def FenwickNodeLo (i : Int) : Int := i - FenwickLowbit i + 1 |
|
|
| def FenwickNodeSum (a : List Int) (i : Int) : Int := sum (sublist (FenwickNodeLo i) (i+1) a) |
|
|
| def FenwickPrefixSum (a : List Int) (pos : Int) : Int := sum (sublist 1 (pos+1) a) |
|
|
| def FenwickAddArray (a : List Int) (pos delta : Int) : List Int := replace_Znth pos (Znth pos a 0+delta) a |
|
|
| def FenwickRep (a bit : List Int) (n : Int) : Prop := |
| Zlength a = n+1 ∧ Zlength bit = n+1 ∧ Znth 0 a 0 = 0 ∧ |
| ∀ i, (1 ≤ i ∧ i ≤ n) → Znth i bit 0 = FenwickNodeSum a i |
|
|
| def FenwickIntervalsIntSafe (a : List Int) (n : Int) : Prop := |
| ∀ lo hi, (1 ≤ lo ∧ lo ≤ hi) → hi ≤ n → |
| -2147483648 ≤ sum (sublist lo (hi+1) a) ∧ sum (sublist lo (hi+1) a) ≤ 2147483647 |
|
|
| def FenwickCovers (node target : Int) : Prop := FenwickNodeLo node ≤ target ∧ target ≤ node |
|
|
| def FenwickAddProgress (entry current : List Int) (n target cursor delta : Int) : Prop := |
| Zlength current = Zlength entry ∧ Znth 0 current 0 = Znth 0 entry 0 ∧ |
| (cursor ≤ n → FenwickCovers cursor target) ∧ ∀ node, (1 ≤ node ∧ node ≤ n) → |
| ((FenwickCovers node target ∧ node < cursor) → Znth node current 0 = Znth node entry 0+delta) ∧ |
| ((¬ FenwickCovers node target ∨ cursor ≤ node) → Znth node current 0 = Znth node entry 0) |
|
|
| def FenwickQueryState (a : List Int) (target cursor accumulator : Int) : Prop := |
| accumulator + FenwickPrefixSum a cursor = FenwickPrefixSum a target |
|
|
| end Data_structures.binary_indexed_tree.lean |
|
|
