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bff4ec7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | 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
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