view redBlackTreeHoare.agda @ 825:02f431665ebc

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 29 Jan 2024 11:42:39 +0900
parents 0b791ae19543
children
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module redBlackTreeHoare where


open import Level hiding (zero)

open import Data.Nat hiding (compare)
open import Data.Nat.Properties as NatProp
open import Data.Maybe
open import Data.Bool hiding ( _<_ )
open import Data.Empty

open import Relation.Binary
open import Relation.Binary.PropositionalEquality

open import stack

record TreeMethods {n m : Level } {a : Set n } {t : Set m } (treeImpl : Set n ) : Set (m Level.⊔ n) where
  field
    putImpl : treeImpl → a → (treeImpl → t) → t
    getImpl  : treeImpl → (treeImpl → Maybe a → t) → t
open TreeMethods

record Tree  {n m : Level } {a : Set n } {t : Set m } (treeImpl : Set n ) : Set (m Level.⊔ n) where
  field
    tree : treeImpl
    treeMethods : TreeMethods {n} {m} {a} {t} treeImpl
  putTree : a → (Tree treeImpl → t) → t
  putTree d next = putImpl (treeMethods ) tree d (\t1 → next (record {tree = t1 ; treeMethods = treeMethods} ))
  getTree : (Tree treeImpl → Maybe a → t) → t
  getTree next = getImpl (treeMethods ) tree (\t1 d → next (record {tree = t1 ; treeMethods = treeMethods} ) d )

open Tree

data Color {n : Level } : Set n where
  Red   : Color
  Black : Color


record Node {n : Level } (a : Set n) (k : ℕ) : Set n where
  inductive
  field
    key   : ℕ
    value : a
    right : Maybe (Node a k)
    left  : Maybe (Node a k)
    color : Color {n}
open Node

record RedBlackTree {n m : Level } {t : Set m} (a : Set n) (k : ℕ) : Set (m Level.⊔ n) where
  field
    root : Maybe (Node a k)
    nodeStack : SingleLinkedStack  (Node a k)
    -- compare : k → k → Tri A B C
    -- <-cmp 

open RedBlackTree

open SingleLinkedStack

compTri : ( x y : ℕ ) ->  Tri ( x < y )  ( x ≡ y )  ( x > y )
compTri = IsStrictTotalOrder.compare (Relation.Binary.StrictTotalOrder.isStrictTotalOrder <-strictTotalOrder)
  where open import  Relation.Binary

-- put new node at parent node, and rebuild tree to the top
--
{-# TERMINATING #-}   -- https://agda.readthedocs.io/en/v2.5.3/language/termination-checking.html
replaceNode : {n m : Level } {t : Set m } {a : Set n} {k : ℕ} → RedBlackTree {n} {m} {t} a k → SingleLinkedStack (Node a k) →  Node a k → (RedBlackTree {n} {m} {t} a k → t) → t
replaceNode {n} {m} {t} {a} {k} tree s n0 next = popSingleLinkedStack s (
      \s parent → replaceNode1 s parent)
       module ReplaceNode where
          replaceNode1 : SingleLinkedStack (Node a k) → Maybe ( Node a k ) → t
          replaceNode1 s nothing = next ( record tree { root = just (record n0 { color = Black}) } )
          replaceNode1 s (just n1) with compTri  (key n1) (key n0)
          replaceNode1 s (just n1) | tri< lt ¬eq ¬gt = replaceNode {n} {m} {t} {a} {k} tree s ( record n1 { value = value n0 ; left = left n0 ; right = right n0 } ) next
          replaceNode1 s (just n1) | tri≈ ¬lt eq ¬gt = replaceNode {n} {m} {t} {a} {k} tree s ( record n1 { left = just n0 } ) next
          replaceNode1 s (just n1) | tri> ¬lt ¬eq gt = replaceNode {n} {m} {t} {a} {k} tree s ( record n1 { right = just n0 } ) next


rotateRight : {n m : Level } {t : Set m } {a : Set n} {k : ℕ} → RedBlackTree {n} {m} {t} a k → SingleLinkedStack (Node  a k) → Maybe (Node a k) → Maybe (Node a k) →
  (RedBlackTree {n} {m} {t} a k → SingleLinkedStack (Node  a k) → Maybe (Node a k) → Maybe (Node a k) → t) → t
rotateRight {n} {m} {t} {a} {k} tree s n0 parent rotateNext = getSingleLinkedStack s (\ s n0 → rotateRight1 {n} {m} {t} {a} {k} tree s n0 parent rotateNext)
  where
        rotateRight1 : {n m : Level } {t : Set m } {a : Set n} {k : ℕ} → RedBlackTree {n} {m} {t} a k → SingleLinkedStack (Node  a k)  → Maybe (Node a k) → Maybe (Node a k) →
          (RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node  a k)  → Maybe (Node a k) → Maybe (Node a k) → t) → t
        rotateRight1 {n} {m} {t} {a} {k}  tree s n0 parent rotateNext with n0
        ... | nothing  = rotateNext tree s nothing n0
        ... | just n1 with parent
        ...           | nothing = rotateNext tree s (just n1 ) n0
        ...           | just parent1 with left parent1
        ...                | nothing = rotateNext tree s (just n1) nothing
        ...                | just leftParent with compTri (key n1) (key leftParent)
        rotateRight1 {n} {m} {t} {a} {k} tree s n0 parent rotateNext | just n1 | just parent1 | just leftParent | tri< a₁ ¬b ¬c = rotateNext tree s (just n1) parent
        rotateRight1 {n} {m} {t} {a} {k} tree s n0 parent rotateNext | just n1 | just parent1 | just leftParent | tri≈ ¬a b ¬c = rotateNext tree s (just n1) parent
        rotateRight1 {n} {m} {t} {a} {k} tree s n0 parent rotateNext | just n1 | just parent1 | just leftParent | tri> ¬a ¬b c = rotateNext tree s (just n1) parent


rotateLeft : {n m  : Level } {t : Set m } {a : Set n} {k : ℕ} → RedBlackTree {n} {m} {t} a k → SingleLinkedStack (Node  a k) → Maybe (Node a k) → Maybe (Node a k) →
  (RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node  a k) → Maybe (Node a k) → Maybe (Node a k) →  t) → t
rotateLeft {n} {m} {t} {a} {k}  tree s n0 parent rotateNext = getSingleLinkedStack s (\ s n0 → rotateLeft1 tree s n0 parent rotateNext)
  where
        rotateLeft1 : {n m  : Level } {t : Set m } {a : Set n} {k : ℕ}  → RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node  a k) → Maybe (Node a k) → Maybe (Node a k) →
          (RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node  a k) → Maybe (Node a k) → Maybe (Node a k) → t) → t
        rotateLeft1 {n} {m} {t} {a} {k}  tree s n0 parent rotateNext with n0
        ... | nothing  = rotateNext tree s nothing n0
        ... | just n1 with parent
        ...           | nothing = rotateNext tree s (just n1) nothing
        ...           | just parent1 with right parent1
        ...                | nothing = rotateNext tree s (just n1) nothing
        ...                | just rightParent with compTri (key n1) (key rightParent)
        rotateLeft1 {n} {m} {t} {a} {k} tree s n0 parent rotateNext | just n1 | just parent1 | just rightParent | tri< a₁ ¬b ¬c = rotateNext tree s (just n1) parent
        rotateLeft1 {n} {m} {t} {a} {k} tree s n0 parent rotateNext | just n1 | just parent1 | just rightParent | tri≈ ¬a b ¬c = rotateNext tree s (just n1) parent
        rotateLeft1 {n} {m} {t} {a} {k} tree s n0 parent rotateNext | just n1 | just parent1 | just rightParent | tri> ¬a ¬b c = rotateNext tree s (just n1) parent
        -- ...                                    | EQ = rotateNext tree s (just n1) parent
        -- ...                                    | _ = rotateNext tree s (just n1) parent

{-# TERMINATING #-}
insertCase5 : {n m  : Level } {t : Set m } {a : Set n} {k : ℕ}  → RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node a k) → Maybe (Node a k) → Node a k → Node a k → (RedBlackTree {n} {m} {t}  a k → t) → t
insertCase5 {n} {m} {t} {a} {k}  tree s n0 parent grandParent next = pop2SingleLinkedStack s (\ s parent grandParent → insertCase51 tree s n0 parent grandParent next)
  where
    insertCase51 : {n m  : Level } {t : Set m } {a : Set n} {k : ℕ}  → RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node a k) → Maybe (Node a k) → Maybe (Node a k) → Maybe (Node a k) → (RedBlackTree {n} {m} {t}  a k → t) → t
    insertCase51 {n} {m} {t} {a} {k}  tree s n0 parent grandParent next with n0
    ...     | nothing = next tree
    ...     | just n1  with  parent | grandParent
    ...                 | nothing | _  = next tree
    ...                 | _ | nothing  = next tree
    ...                 | just parent1 | just grandParent1 with left parent1 | left grandParent1
    ...                                                     | nothing | _  = next tree
    ...                                                     | _ | nothing  = next tree
    ...                                                     | just leftParent1 | just leftGrandParent1
      with compTri (key n1) (key leftParent1) | compTri (key leftParent1) (key leftGrandParent1)
    ...    | tri≈ ¬a b ¬c | tri≈ ¬a1 b1 ¬c1  = rotateRight tree s n0 parent (\ tree s n0 parent → insertCase5 tree s n0 parent1 grandParent1 next)
    ...    | _            | _                = rotateLeft tree s n0 parent (\ tree s n0 parent → insertCase5 tree s n0 parent1 grandParent1 next)
    -- ...     | EQ | EQ = rotateRight tree s n0 parent (\ tree s n0 parent → insertCase5 tree s n0 parent1 grandParent1 next)
    -- ...     | _ | _ = rotateLeft tree s n0 parent (\ tree s n0 parent → insertCase5 tree s n0 parent1 grandParent1 next)

insertCase4 : {n m  : Level } {t : Set m } {a : Set n} {k : ℕ}  → RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node a k) → Node a k → Node a k → Node a k → (RedBlackTree {n} {m} {t}  a k → t) → t
insertCase4 {n} {m} {t} {a} {k}  tree s n0 parent grandParent next
       with  (right parent) | (left grandParent)
...    | nothing | _ = insertCase5 tree s (just n0) parent grandParent next
...    | _ | nothing = insertCase5 tree s (just n0) parent grandParent next
...    | just rightParent | just leftGrandParent with compTri (key n0) (key rightParent) | compTri (key parent) (key leftGrandParent) -- (key n0) (key rightParent) | (key parent) (key leftGrandParent)
-- ...                                              | EQ | EQ = popSingleLinkedStack s (\ s n1 → rotateLeft tree s (left n0) (just grandParent)
--    (\ tree s n0 parent → insertCase5 tree s n0 rightParent grandParent next))
-- ...                                              | _ | _  = insertCase41 tree s n0 parent grandParent next
...                                                 | tri≈ ¬a b ¬c | tri≈ ¬a1 b1 ¬c1 = popSingleLinkedStack s (\ s n1 → rotateLeft tree s (left n0) (just grandParent) (\ tree s n0 parent → insertCase5 tree s n0 rightParent grandParent next))
... | _            | _               = insertCase41 tree s n0 parent grandParent next
  where
    insertCase41 : {n m  : Level } {t : Set m } {a : Set n} {k : ℕ}  → RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node a k) → Node a k → Node a k → Node a k → (RedBlackTree {n} {m} {t}  a k → t) → t
    insertCase41 {n} {m} {t} {a} {k}  tree s n0 parent grandParent next
                 with  (left parent) | (right grandParent)
    ...    | nothing | _ = insertCase5 tree s (just n0) parent grandParent next
    ...    | _ | nothing = insertCase5 tree s (just n0) parent grandParent next
    ...    | just leftParent | just rightGrandParent with compTri (key n0) (key leftParent) | compTri (key parent) (key rightGrandParent)
    ... | tri≈ ¬a b ¬c | tri≈ ¬a1 b1 ¬c1 =  popSingleLinkedStack s (\ s n1 → rotateRight tree s (right n0) (just grandParent) (\ tree s n0 parent → insertCase5 tree s n0 leftParent grandParent next))
    ... | _ | _ = insertCase5 tree s (just n0) parent grandParent next
    -- ...                                              | EQ | EQ = popSingleLinkedStack s (\ s n1 → rotateRight tree s (right n0) (just grandParent)
    --    (\ tree s n0 parent → insertCase5 tree s n0 leftParent grandParent next))
    -- ...                                              | _ | _  = insertCase5 tree s (just n0) parent grandParent next

colorNode : {n : Level } {a : Set n} {k : ℕ} → Node a k → Color  → Node a k
colorNode old c = record old { color = c }

{-# TERMINATING #-}
insertNode : {n m  : Level } {t : Set m } {a : Set n} {k : ℕ}  → RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node a k) → Node a k → (RedBlackTree {n} {m} {t}  a k → t) → t
insertNode {n} {m} {t} {a} {k}  tree s n0 next = get2SingleLinkedStack s (insertCase1 n0)
   where
    insertCase1 : Node a k → SingleLinkedStack (Node a k) → Maybe (Node a k) → Maybe (Node a k) → t    -- placed here to allow mutual recursion
          -- http://agda.readthedocs.io/en/v2.5.2/language/mutual-recursion.html
    insertCase3 : SingleLinkedStack (Node a k) → Node a k → Node a k → Node a k → t
    insertCase3 s n0 parent grandParent with left grandParent | right grandParent
    ... | nothing | nothing = insertCase4 tree s n0 parent grandParent next
    ... | nothing | just uncle  = insertCase4 tree s n0 parent grandParent next
    ... | just uncle | _  with compTri ( key uncle ) ( key parent )
    insertCase3 s n0 parent grandParent | just uncle | _ | tri≈ ¬a b ¬c = insertCase4 tree s n0 parent grandParent next
    insertCase3 s n0 parent grandParent | just uncle | _ | tri< a ¬b ¬c with color uncle
    insertCase3 s n0 parent grandParent | just uncle | _ | tri< a ¬b ¬c | Red = pop2SingleLinkedStack s ( \s p0 p1 → insertCase1  (
           record grandParent { color = Red ; left = just ( record parent { color = Black } )  ; right = just ( record uncle { color = Black } ) }) s p0 p1 )
    insertCase3 s n0 parent grandParent | just uncle | _ | tri< a ¬b ¬c | Black = insertCase4 tree s n0 parent grandParent next
    insertCase3 s n0 parent grandParent | just uncle | _ | tri> ¬a ¬b c with color uncle
    insertCase3 s n0 parent grandParent | just uncle | _ | tri> ¬a ¬b c | Red = pop2SingleLinkedStack s ( \s p0 p1 → insertCase1  ( record grandParent { color = Red ; left = just ( record parent { color = Black } )  ; right = just ( record uncle { color = Black } ) }) s p0 p1 )
    insertCase3 s n0 parent grandParent | just uncle | _ | tri> ¬a ¬b c | Black = insertCase4 tree s n0 parent grandParent next
    -- ...                   | EQ =  insertCase4 tree s n0 parent grandParent next
    -- ...                   | _ with color uncle
    -- ...                           | Red = pop2SingleLinkedStack s ( \s p0 p1 → insertCase1  (
    --        record grandParent { color = Red ; left = just ( record parent { color = Black } )  ; right = just ( record uncle { color = Black } ) }) s p0 p1 )
    -- ...                           | Black = insertCase4 tree s n0 parent grandParent next --!!
    insertCase2 : SingleLinkedStack (Node a k) → Node a k → Node a k → Node a k → t
    insertCase2 s n0 parent grandParent with color parent
    ... | Black = replaceNode tree s n0 next
    ... | Red =   insertCase3 s n0 parent grandParent
    insertCase1 n0 s nothing nothing = next tree
    insertCase1 n0 s nothing (just grandParent) = next tree
    insertCase1 n0 s (just parent) nothing = replaceNode tree s (colorNode n0 Black) next
    insertCase1 n0 s (just parent) (just grandParent) = insertCase2 s n0 parent grandParent


----
-- find node potition to insert or to delete, the path will be in the stack
--
findNode : {n m  : Level } {a : Set n} {k : ℕ} {t : Set m}  → RedBlackTree {n} {m} {t}   a k → SingleLinkedStack (Node a k) → (Node a k) → (Node a k) → (RedBlackTree {n} {m} {t}  a k → SingleLinkedStack (Node a k) → Node a k → t) → t
findNode {n} {m} {a} {k} {t} tree s n0 n1 next = pushSingleLinkedStack s n1 (\ s → findNode1 s n1)
  module FindNode where
    findNode2 : SingleLinkedStack (Node a k) → (Maybe (Node a k)) → t
    findNode2 s nothing = next tree s n0
    findNode2 s (just n) = findNode tree s n0 n next
    findNode1 : SingleLinkedStack (Node a k) → (Node a k)  → t
    findNode1 s n1 with (compTri (key n0) (key n1))
    findNode1 s n1 | tri< a ¬b ¬c = popSingleLinkedStack s ( \s _ → next tree s (record n1 { key = key n1 ; value = value n0 } ) )
    findNode1 s n1 | tri≈ ¬a b ¬c = findNode2 s (right n1)
    findNode1 s n1 | tri> ¬a ¬b c = findNode2 s (left n1)
    -- ...                                | EQ = popSingleLinkedStack s ( \s _ → next tree s (record n1 { key = key n1 ; value = value n0 } ) )
    -- ...                                | GT = findNode2 s (right n1)
    -- ...                                | LT = findNode2 s (left n1)




leafNode : {n : Level } { a : Set n } → a → (k : ℕ) → (Node a k)
leafNode v k1 = record { key = k1 ; value = v ; right = nothing ; left = nothing ; color = Red }

putRedBlackTree : {n m : Level} {t : Set m} {a : Set n} {k : ℕ} → RedBlackTree {n} {m} {t} a k → {!!} → {!!} → (RedBlackTree {n} {m} {t} a k → t) → t
putRedBlackTree {n} {m} {t} {a} {k} tree val k1 next with (root tree)
putRedBlackTree {n} {m} {t} {a} {k} tree val k1 next | nothing = next (record tree {root = just (leafNode {!!} {!!}) })
putRedBlackTree {n} {m} {t} {a} {k} tree val k1 next | just n2 = clearSingleLinkedStack (nodeStack tree) (λ s → findNode tree s (leafNode {!!} {!!}) n2 (λ tree1 s n1 → insertNode tree1 s n1 next))
-- putRedBlackTree {n} {m} {t} {a} {k} tree value k1 next with (root tree)
-- ...                                | nothing = next (record tree {root = just (leafNode k1 value) })
-- ...                                | just n2  = clearSingleLinkedStack (nodeStack tree) (\ s → findNode tree s (leafNode k1 value) n2 (\ tree1 s n1 → insertNode tree1 s n1 next))


-- getRedBlackTree : {n m  : Level } {t : Set m}  {a : Set n} {k : ℕ} → RedBlackTree {n} {m} {t} {A}  a k → k → (RedBlackTree {n} {m} {t} {A}  a k → (Maybe (Node a k)) → t) → t
-- getRedBlackTree {_} {_} {t}  {a} {k} tree k1 cs = checkNode (root tree)
--   module GetRedBlackTree where                     -- http://agda.readthedocs.io/en/v2.5.2/language/let-and-where.html
--     search : Node a k → t
--     checkNode : Maybe (Node a k) → t
--     checkNode nothing = cs tree nothing
--     checkNode (just n) = search n
--     search n with compTri k1 (key n)
--     search n | tri< a ¬b ¬c = checkNode (left n)
--     search n | tri≈ ¬a b ¬c = cs tree (just n)
--     search n | tri> ¬a ¬b c = checkNode (right n)



-- compareT :  {A B C : Set } → ℕ → ℕ → Tri A B C
-- compareT x y with IsStrictTotalOrder.compare (Relation.Binary.StrictTotalOrder.isStrictTotalOrder <-strictTotalOrder) x y
-- compareT x y | tri< a ¬b ¬c = tri< {!!} {!!} {!!}
-- compareT x y | tri≈ ¬a b ¬c = {!!}
-- compareT x y | tri> ¬a ¬b c = {!!}
-- -- ... | tri≈ a b c = {!!}
-- -- ... | tri< a b c = {!!}
-- -- ... | tri> a b c = {!!}

-- compare2 : (x y : ℕ ) → CompareResult {Level.zero}
-- compare2 zero zero = EQ
-- compare2 (suc _) zero = GT
-- compare2  zero (suc _) = LT
-- compare2  (suc x) (suc y) = compare2 x y

-- -- putUnblanceTree : {n m : Level } {a : Set n} {k : ℕ} {t : Set m} → RedBlackTree {n} {m} {t} {A}  a k → k → a → (RedBlackTree {n} {m} {t} {A}  a k → t) → t
-- -- putUnblanceTree {n} {m} {A} {a} {k} {t} tree k1 value next with (root tree)
-- -- ...                                | nothing = next (record tree {root = just (leafNode k1 value) })
-- -- ...                                | just n2  = clearSingleLinkedStack (nodeStack tree) (λ  s → findNode tree s (leafNode k1 value) n2 (λ  tree1 s n1 → replaceNode tree1 s n1 next))

-- -- checkT : {m : Level } (n : Maybe (Node  ℕ ℕ)) → ℕ → Bool
-- -- checkT nothing _ = false
-- -- checkT (just n) x with compTri (value n)  x
-- -- ...  | tri≈ _ _ _ = true
-- -- ...  | _ = false

-- -- checkEQ :  {m : Level }  ( x :  ℕ ) -> ( n : Node  ℕ ℕ ) -> (value n )  ≡ x  -> checkT {m} (just n) x ≡ true
-- -- checkEQ x n refl with compTri (value n)  x
-- -- ... |  tri≈ _ refl _ = refl
-- -- ... |  tri> _ neq gt =  ⊥-elim (neq refl)
-- -- ... |  tri< lt neq _ =  ⊥-elim (neq refl)


createEmptyRedBlackTreeℕ : {n m  : Level} {t : Set m} (a : Set n) (b : ℕ)
     → RedBlackTree {n} {m} {t} a b
createEmptyRedBlackTreeℕ a b = record {
        root = nothing
     ;  nodeStack = emptySingleLinkedStack
     -- ;  nodeComp = λ x x₁ → {!!}

   }

-- ( x y : ℕ ) ->  Tri  ( x < y )  ( x ≡ y )  ( x > y )

-- test = (λ x → (createEmptyRedBlackTreeℕ x x) 

-- ts = createEmptyRedBlackTreeℕ {ℕ} {?} {!!} 0

-- tes = putRedBlackTree {_} {_} {_} (createEmptyRedBlackTreeℕ {_} {_} {_} 3 3) 2 2 (λ t → t)