worked on chapter 5

2026-03-17 11:38:06 +01:00
parent 61da31dcdc
commit 93747eaf7a
7 changed files with 406 additions and 0 deletions
--- a/chap_5/e5_7_2.dfy
+++ b/chap_5/e5_7_2.dfy
@@ -0,0 +1,134 @@
+datatype BYTree = BlueLeaf | YellowLeaf | Node(left: BYTree, right: BYTree)
+
+function ReverseColors(t: BYTree): BYTree {
+    match t
+    case BlueLeaf => YellowLeaf
+    case YellowLeaf => BlueLeaf 
+    case Node(left, right) => BYTree.Node(ReverseColors(left), ReverseColors(right))
+}
+
+lemma {:induction false} ReverseColorsInvolution(t: BYTree)
+    ensures ReverseColors(ReverseColors(t)) == t
+{     
+    match t
+    case BlueLeaf =>
+    case YellowLeaf =>
+    case Node(left, right) =>
+        calc {
+            ReverseColors(ReverseColors(Node(left,right)));
+        ==
+            ReverseColors(Node(ReverseColors(left), ReverseColors(right)));
+        ==  
+            Node(ReverseColors(ReverseColors(left)), ReverseColors(ReverseColors(right)));
+        ==  {ReverseColorsInvolution(left); ReverseColorsInvolution(right);}
+            Node(left, right);
+        }
+}
+
+function LeafCount(t: BYTree): nat 
+{
+    match t
+    case BlueLeaf | YellowLeaf => 1
+    case Node(left, right) => LeafCount(left) + LeafCount(right)
+}
+
+lemma {:induction false} ReversePreserveLeafCount(t: BYTree) 
+    ensures LeafCount(ReverseColors(t)) == LeafCount(t)
+    decreases t
+{
+    match t
+    case BlueLeaf =>
+    case YellowLeaf =>
+    case Node(left, right) => 
+        calc {
+            LeafCount(ReverseColors(Node(left, right)));
+        ==
+            LeafCount(Node(ReverseColors(left), ReverseColors(right)));
+        ==
+            LeafCount(ReverseColors(left)) + LeafCount(ReverseColors(right));
+        ==  {ReversePreserveLeafCount(left); ReversePreserveLeafCount(right);}
+            LeafCount(left) + LeafCount(right);
+        ==
+            LeafCount(Node(left, right));
+        }
+}
+
+function Oceanize(t: BYTree): BYTree {
+    match t
+        case BlueLeaf | YellowLeaf => BlueLeaf
+        case Node(left, right) => Node(Oceanize(left), Oceanize(right))
+}
+
+lemma {:induction false} IdempotentOceanize(t: BYTree) 
+    ensures Oceanize(t) == Oceanize(Oceanize(t))
+    decreases t
+{
+    match t
+    case BlueLeaf => 
+        calc {
+            Oceanize(Oceanize(BlueLeaf));
+        ==
+            Oceanize(BlueLeaf);
+        ==
+            BlueLeaf;
+        }
+    case YellowLeaf =>
+        calc {
+            Oceanize(Oceanize(YellowLeaf));
+        ==
+            Oceanize(BlueLeaf);
+        ==
+            BlueLeaf;
+        ==
+            Oceanize(YellowLeaf);
+        }
+        
+    case Node(left, right) =>
+        calc {
+            Oceanize(Node(left, right));
+        ==
+            Node(Oceanize(left), Oceanize(right));
+        ==  {IdempotentOceanize(left); IdempotentOceanize(right);}
+            Node(Oceanize(Oceanize(left)), Oceanize(Oceanize(right)));
+        ==
+            Oceanize(Node(Oceanize(left), Oceanize(right)));
+        ==
+            Oceanize(Oceanize(Node(left,right)));
+        }
+}
+
+function BlueCount(t: BYTree): nat {
+    match t
+    case BlueLeaf => 1
+    case YellowLeaf => 0
+    case Node(left, right) =>
+        BlueCount(left) + BlueCount(right)
+}
+
+lemma {:induction false} BlueCountBound(t: BYTree) 
+    ensures BlueCount(t) <= BlueCount(Oceanize(t))
+{
+    match t
+    case BlueLeaf => 
+        calc {
+            BlueCount(BlueLeaf);
+        ==
+            BlueCount(Oceanize(BlueLeaf));
+        }
+    case YellowLeaf =>
+        calc {
+            BlueCount(YellowLeaf);
+        <
+            BlueCount(BlueLeaf);
+        ==
+            BlueCount(Oceanize(YellowLeaf));
+        }
+    case Node(left, right) =>
+        calc {
+            BlueCount(Node(left, right));
+        ==
+            BlueCount(left) + BlueCount(right);
+        <= {BlueCountBound(left); BlueCountBound(right);}
+            BlueCount(Oceanize(left)) + BlueCount(Oceanize(right));
+        }
+}