diff options
Diffstat (limited to 'fp1/week7')
| -rw-r--r-- | fp1/week7/camil/BewijsMeppenEnTippen.icl | 82 | ||||
| -rw-r--r-- | fp1/week7/camil/BinSearchTree.dcl | 7 | ||||
| -rw-r--r-- | fp1/week7/camil/BinSearchTree.icl | 177 | ||||
| -rw-r--r-- | fp1/week7/camil/BinTree.dcl | 16 | ||||
| -rw-r--r-- | fp1/week7/camil/BinTree.icl | 38 | ||||
| -rw-r--r-- | fp1/week7/mart/BewijsMeppenEnTippen.icl | 29 | ||||
| -rw-r--r-- | fp1/week7/mart/BinSearchTree.dcl | 7 | ||||
| -rw-r--r-- | fp1/week7/mart/BinSearchTree.icl | 141 |
8 files changed, 497 insertions, 0 deletions
diff --git a/fp1/week7/camil/BewijsMeppenEnTippen.icl b/fp1/week7/camil/BewijsMeppenEnTippen.icl new file mode 100644 index 0000000..ca5e396 --- /dev/null +++ b/fp1/week7/camil/BewijsMeppenEnTippen.icl @@ -0,0 +1,82 @@ +// Mart Lubbers, s4109503
+// Camil Staps, s4498062
+
+Zij gegeven:
+
+:: BTree a = Tip a | Bin (BTree a) (BTree a)
+
+map :: (a -> b) [a] -> [b]
+map f [] = [] (1.)
+map f [x:xs] = [f x : map f xs] (2.)
+
+mapbtree :: (a -> b) (BTree a) -> BTree b
+mapbtree f (Tip a) = Tip (f a) (3.)
+mapbtree f (Bin t1 t2) = Bin (mapbtree f t1) (mapbtree f t2) (4.)
+
+foldbtree :: (a a -> a) (BTree a) -> a
+foldbtree f (Tip x) = x (5.)
+foldbtree f (Bin t1 t2) = f (foldbtree f t1) (foldbtree f t2) (6.)
+
+tips :: (BTree a) -> [a]
+tips t = foldbtree (++) (mapbtree unit t) (7.)
+
+unit :: a -> [a]
+unit x = [x] (8.)
+
+
+Te bewijzen:
+ voor alle functies f, voor alle eindige bomen t:
+
+ map f (tips t) = tips (mapbtree f t)
+
+Bewijs:
+ Met inductie over t.
+
+ Inductiebasis: stel t = Tip a.
+ Dan hebben we:
+
+ map f (tips t) // definitie tips (7)
+ = map f (foldbtree (++) (mapbtree unit t)) // aanname t = Tip a
+ = map f (foldbtree (++) (mapbtree unit (Tip a))) // definitie mapbtree (3)
+ = map f (foldbtree (++) (Tip unit a)) // definitie foldbtree (5)
+ = map f (unit a) // definitie unit (8)
+ = map f [a] // herschrijven lijst
+ = map f [a:[]] // definitie map (2)
+ = [f a : map f []] // definitie map (1)
+ = [f a : []] // herschrijven lijst
+ = [f a] // definitie unit (8)
+ = unit (f a) // definitie foldbtree (5)
+ = foldbtree (++) (Tip (unit (f a))) // definitie mapbtree (3)
+ = foldbtree (++) (mapbtree unit (Tip (f a))) // definitie tips (7)
+ = tips (Tip (f a)) // definitie mapbtree (3)
+ = tips (mapbtree f (Tip a)) // aanname t = Tip a
+ = tips (mapbtree f t)
+
+ Dus de stelling geldt voor t = Tip a.
+
+ Inductiestap: laten we aannemen dat
+ map f (tips t) = tips (mapbtree f t)
+ voor alle f en zekere t=t1,t=t2 (inductiehypothese).
+ Dan hebben we:
+
+ map f (tips (Bin t1 t2)) // definitie tips (7)
+ = map f (foldbtree (++) (mapbtree unit (Bin t1 t2))) // definitie mapbtree (4)
+ = map f (foldbtree (++) (Bin (mapbtree unit t1) (mapbtree unit t2))) // definitie foldbtree (6)
+ = map f ((++) (foldbtree (++) (mapbtree unit t1)) (foldbtree (++) (mapbtree unit t2))) // definitie tips (7)
+ = map f ((++) (tips t1) (tips t2)) // 9.4.1
+ = (map f (tips t1)) ++ (map f (tips t2)) // inductiehypothese
+ = (tips (mapbtree f t1)) ++ (tips (mapbtree f t2)) // definitie tips (7)
+ = (foldbtree (++) (mapbtree unit (f t1))) ++ (foldbtree (++) (mapbtree unit (f t2))) // herschrijven infixnotatie
+ = (++) (foldbtree (++) (mapbtree unit (f t1))) (foldbtree (++) (mapbtree unit (f t2))) // definitie foldbtree (6)
+ = foldbtree (++) (Bin (mapbtree unit (f t1)) (mapbtree unit (f t2))) // definitie mapbtree (4)
+ = foldbtree (++) (mapbtree unit (Bin (mapbtree f t1) (mapbtree f t2))) // definitie tips (7)
+ = tips (Bin (mapbtree f t1) (mapbtree f t2)) // definitie mapbtree (4)
+ = tips (mapbtree f (Bin t1 t2))
+
+ Conclusie:
+ We hebben laten zien dat de stelling geldt voor elke f met t = Tip a. Vervolgens hebben we laten zien dat als de stelling geldt voor elke f met t=t1 of t=t2, de stelling óók geldt voor elke f met t = Bin t1 t2.
+ Met het principe van inductie volgt nu
+
+ map f (tips t) = tips (mapbtree f t)
+
+ voor alle functies f en alle eindige bomen t.
\ No newline at end of file diff --git a/fp1/week7/camil/BinSearchTree.dcl b/fp1/week7/camil/BinSearchTree.dcl new file mode 100644 index 0000000..696b065 --- /dev/null +++ b/fp1/week7/camil/BinSearchTree.dcl @@ -0,0 +1,7 @@ +definition module BinSearchTree
+
+import StdClass
+import BinTree
+
+is_geordend :: (Tree a) -> Bool | Ord a // meest algemene type
+is_gebalanceerd :: (Tree a) -> Bool | Ord a // meest algemene type
diff --git a/fp1/week7/camil/BinSearchTree.icl b/fp1/week7/camil/BinSearchTree.icl new file mode 100644 index 0000000..559846b --- /dev/null +++ b/fp1/week7/camil/BinSearchTree.icl @@ -0,0 +1,177 @@ +// Mart Lubbers, s4109503 +// Camil Staps, s4498062 + +implementation module BinSearchTree + +import StdEnv +import BinTree + +z0 = Leaf +// Leaf + +z1 = insertTree 50 z0 +// 50 +// | +// ------------- +// | | +// Leaf Leaf + +z2 = insertTree 10 z1 +// 50 +// | +// ------------- +// | | +// 10 Leaf +// | +// --------- +// | | +// Leaf Leaf + +z3 = insertTree 75 z2 +// 50 +// | +// --------------- +// | | +// 10 75 +// | | +// --------- --------- +// | | | | +// Leaf Leaf Leaf Leaf + +z4 = insertTree 80 z3 +// 50 +// | +// --------------- +// | | +// 10 75 +// | | +// --------- --------- +// | | | | +// Leaf Leaf Leaf 80 +// | +// --------- +// | | +// Leaf Leaf + +z5 = insertTree 77 z4 +// 50 +// | +// --------------- +// | | +// 10 75 +// | | +// --------- --------- +// | | | | +// Leaf Leaf Leaf 77 +// | +// --------- +// | | +// Leaf 80 +// | +// --------- +// | | +// Leaf Leaf + +z6 = insertTree 10 z5 +// 50 +// | +// --------------- +// | | +// 10 75 +// | | +// --------- --------- +// | | | | +// 10 Leaf Leaf 77 +// | | +// --------- --------- +// | | | | +// Leaf Leaf Leaf 80 +// | +// --------- +// | | +// Leaf Leaf + +z7 = insertTree 75 z6 +// 50 +// | +// ---------------- +// | | +// 10 75 +// | | +// --------- ----------- +// | | | | +// 10 Leaf 75 77 +// | | | +// --------- ------ ------- +// | | | | | | +// Leaf Leaf Leaf Leaf Leaf 80 +// | +// --------- +// | | +// Leaf Leaf + +z8 = deleteTree 50 z7 +// 10 +// | +// ---------------- +// | | +// 10 75 +// | | +// --------- ----------- +// | | | | +// Leaf Leaf 75 77 +// | | +// ------ ------- +// | | | | +// Leaf Leaf Leaf 80 +// | +// --------- +// | | +// Leaf Leaf + +// Uit het diktaat, blz. 73: +insertTree :: a (Tree a) -> Tree a | Ord a +insertTree e Leaf = Node e Leaf Leaf +insertTree e (Node x le ri) +| e <= x = Node x (insertTree e le) ri +| e > x = Node x le (insertTree e ri) + +deleteTree :: a (Tree a) -> (Tree a) | Eq, Ord a +deleteTree e Leaf = Leaf +deleteTree e (Node x le ri) +| e < x = Node x (deleteTree e le) ri +| e == x = join le ri +| e > x = Node x le (deleteTree e ri) +where + join :: (Tree a) (Tree a) -> (Tree a) + join Leaf b2 = b2 + join b1 b2 = Node x b1` b2 + where + (x,b1`) = largest b1 + + largest :: (Tree a) -> (a,(Tree a)) + largest (Node x b1 Leaf) = (x,b1) + largest (Node x b1 b2) = (y,Node x b1 b2`) + where + (y,b2`) = largest b2 + + +is_geordend :: (Tree a) -> Bool | Ord a // meest algemene type +is_geordend Leaf = True +is_geordend (Node x le ri) = (foldr (&&) True (map ((>) x) (members le))) && (foldr (&&) True (map ((<=) x) (members ri))) && is_geordend le && is_geordend ri +where + members :: (Tree a) -> [a] + members Leaf = [] + members (Node x le ri) = [x:(members le) ++ (members ri)] + +//Start = map is_geordend [t0,t1,t2,t3,t4,t5,t6,t7] + +is_gebalanceerd :: (Tree a) -> Bool | Ord a // meest algemene type +is_gebalanceerd Leaf = True +is_gebalanceerd (Node x le ri) = abs ((depth le) - (depth ri)) <= 1 && is_gebalanceerd le && is_gebalanceerd ri +where + depth :: (Tree a) -> Int + depth Leaf = 0 + depth (Node x le ri) = max (depth le) (depth ri) + 1 + +//Start = map is_gebalanceerd [t0,t1,t2,t3,t4,t5,t6,t7] diff --git a/fp1/week7/camil/BinTree.dcl b/fp1/week7/camil/BinTree.dcl new file mode 100644 index 0000000..7774ece --- /dev/null +++ b/fp1/week7/camil/BinTree.dcl @@ -0,0 +1,16 @@ +definition module BinTree
+
+:: Tree a = Node a (Tree a) (Tree a) | Leaf
+
+t0 :: Tree Int
+t1 :: Tree Int
+t2 :: Tree Int
+t3 :: Tree Int
+t4 :: Tree Int
+t5 :: Tree Int
+t6 :: Tree Int
+t7 :: Tree Int
+
+//nodes :: // meest algemene type
+//leaves :: // meest algemene type
+//diepte :: // meest algemene type
diff --git a/fp1/week7/camil/BinTree.icl b/fp1/week7/camil/BinTree.icl new file mode 100644 index 0000000..601efcc --- /dev/null +++ b/fp1/week7/camil/BinTree.icl @@ -0,0 +1,38 @@ +implementation module BinTree
+
+import StdEnv
+
+:: Tree a = Node a (Tree a) (Tree a) | Leaf
+
+t0 :: Tree Int
+t0 = Leaf
+t1 :: Tree Int
+t1 = Node 4 t0 t0
+t2 :: Tree Int
+t2 = Node 2 t0 t1
+t3 :: Tree Int
+t3 = Node 5 t2 t0
+t4 :: Tree Int
+t4 = Node 5 t2 t2
+t5 :: Tree Int
+t5 = Node 1 Leaf (Node 2 Leaf (Node 3 Leaf (Node 4 Leaf Leaf)))
+t6 :: Tree Int
+t6 = Node 1 (Node 2 (Node 3 (Node 4 Leaf Leaf) Leaf) Leaf) Leaf
+t7 :: Tree Int
+t7 = Node 4 (Node 1 Leaf Leaf) (Node 5 (Node 2 Leaf Leaf) Leaf)
+
+// 2.
+//nodes :: // meest algemene type
+//nodes ...
+
+//Start = map nodes [t0,t1,t2,t3,t4,t5,t6,t7]
+
+//leaves :: // meest algemene type
+//leaves ...
+
+//Start = map leaves [t0,t1,t2,t3,t4,t5,t6,t7]
+
+//diepte :: // meest algemene type
+//diepte ...
+
+//Start = map diepte [t0,t1,t2,t3,t4,t5,t6,t7]
diff --git a/fp1/week7/mart/BewijsMeppenEnTippen.icl b/fp1/week7/mart/BewijsMeppenEnTippen.icl new file mode 100644 index 0000000..720ff4d --- /dev/null +++ b/fp1/week7/mart/BewijsMeppenEnTippen.icl @@ -0,0 +1,29 @@ +Zij gegeven:
+
+:: BTree a = Tip a | Bin (BTree a) (BTree a)
+
+map :: (a -> b) [a] -> [b]
+map f [] = [] (1.)
+map f [x:xs] = [f x : map f xs] (2.)
+
+mapbtree :: (a -> b) (BTree a) -> BTree b
+mapbtree f (Tip a) = Tip (f a) (3.)
+mapbtree f (Bin t1 t2) = Bin (mapbtree f t1) (mapbtree f t2) (4.)
+
+foldbtree :: (a a -> a) (BTree a) -> a
+foldbtree f (Tip x) = x (5.)
+foldbtree f (Bin t1 t2) = f (foldbtree f t1) (foldbtree f t2) (6.)
+
+tips :: (BTree a) -> [a]
+tips t = foldbtree (++) (mapbtree unit t) (7.)
+
+unit :: a -> [a]
+unit x = [x] (8.)
+
+
+Te bewijzen:
+ voor alle functies f, voor alle eindige bomen t:
+
+ map f (tips t) = tips (mapbtree f t)
+
+Bewijs:
diff --git a/fp1/week7/mart/BinSearchTree.dcl b/fp1/week7/mart/BinSearchTree.dcl new file mode 100644 index 0000000..2e480bb --- /dev/null +++ b/fp1/week7/mart/BinSearchTree.dcl @@ -0,0 +1,7 @@ +definition module BinSearchTree
+
+import StdClass
+import BinTree
+
+is_geordend :: // meest algemene type
+is_gebalanceerd :: // meest algemene type
diff --git a/fp1/week7/mart/BinSearchTree.icl b/fp1/week7/mart/BinSearchTree.icl new file mode 100644 index 0000000..8f9f05c --- /dev/null +++ b/fp1/week7/mart/BinSearchTree.icl @@ -0,0 +1,141 @@ +implementation module BinSearchTree
+
+import StdEnv
+import BinTree
+
+
+z0
+ Leaf
+z1
+ 50
+ |
+ ----------
+ | |
+ Leaf Leaf
+z2
+ 50
+ |
+ ----------
+ | |
+ 10 Leaf
+ |
+ ------
+ | |
+ Leaf Leaf
+z3
+ 50
+ |
+ ----------
+ | |
+ 10 75
+ | |
+ ------ ------
+ | | | |
+ Leaf Leaf Leaf Leaf
+z4
+ 50
+ |
+ ----------
+ | |
+ 10 75
+ | |
+ ------ ------
+ | | | |
+ Leaf Leaf Leaf 80
+ |
+ ------
+ | |
+ Leaf Leaf
+z5
+ 50
+ |
+ ----------
+ | |
+ 10 75
+ | |
+ ------ ------
+ | | | |
+ Leaf Leaf Leaf 77
+ |
+ ------
+ | |
+ Leaf 80
+ |
+ ------
+ | |
+ Leaf Leaf
+z6
+ 50
+ |
+ ----------
+ | |
+ 10 75
+ | |
+ ------ ------
+ | | | |
+ 10 Leaf Leaf 77
+ | |
+ ------ ------
+ | | | |
+Leaf Leaf Leaf 80
+ |
+ ------
+ | |
+ Leaf Leaf
+z7
+ 50
+ |
+ ----------
+ | |
+ 10 75
+ | |
+ ------ -----------
+ | | | |
+ 10 Leaf 75 77
+ | | |
+ ------ ------ ------
+ | | | | | |
+Leaf Leaf Leaf Leaf Leaf 80
+ |
+ ------
+ | |
+ Leaf Leaf
+z8
+
+// Uit het diktaat, blz. 73:
+insertTree :: a (Tree a) -> Tree a | Ord a
+insertTree e Leaf = Node e Leaf Leaf
+insertTree e (Node x le ri)
+| e <= x = Node x (insertTree e le) ri
+| e > x = Node x le (insertTree e ri)
+
+deleteTree :: a (Tree a) -> (Tree a) | Eq, Ord a
+deleteTree e Leaf = Leaf
+deleteTree e (Node x le ri)
+| e < x = Node x (deleteTree e le) ri
+| e == x = join le ri
+| e > x = Node x le (deleteTree e ri)
+where
+ join :: (Tree a) (Tree a) -> (Tree a)
+ join Leaf b2 = b2
+ join b1 b2 = Node x b1` b2
+ where
+ (x,b1`) = largest b1
+
+ largest :: (Tree a) -> (a,(Tree a))
+ largest (Node x b1 Leaf) = (x,b1)
+ largest (Node x b1 b2) = (y,Node x b1 b2`)
+ where
+ (y,b2`) = largest b2
+
+
+is_geordend :: // meest algemene type
+is_geordend ...
+
+Start = map is_geordend [t0,t1,t2,t3,t4,t5,t6,t7]
+
+
+is_gebalanceerd :: // meest algemene type
+is_gebalanceerd ...
+
+//Start = map is_gebalanceerd [t0,t1,t2,t3,t4,t5,t6,t7]
|