Mercurial > hg > Gears > GearsAgda
view redBlackTreeHoare.agda @ 832:01658441313e
RTtoTI0 done
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 14 Mar 2024 18:14:41 +0900 |
| parents | 0b791ae19543 |
| children |
line wrap: on
line source
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)