Mercurial > hg > Members > Moririn
diff redBlackTreeHoare.agda @ 575:73fc32092b64
push local rbtree
| author | ryokka |
|---|---|
| date | Fri, 01 Nov 2019 17:42:51 +0900 |
| parents | |
| children | 40d01b368e34 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/redBlackTreeHoare.agda Fri Nov 01 17:42:51 2019 +0900 @@ -0,0 +1,290 @@ +module RedBlackTree 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 +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 + +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)