Finished Chapter 4

chap_3/e3_2.dfy

@@ -0,0 +1,17 @@
+function M(x: int, b: bool): int 
+    decreases !b
+{
+    if b then x else M(x + 25, true) 
+}
+
+function N(x: int, y: int, b: bool): int 
+    decreases x, b
+{
+    if x <= 0 || y <= 0 then
+        x + y
+    else if b then
+        N(x, y + 3, !b)
+    else
+        N(x - 1, y, true)
+}
+

chap_3/e3_3.dfy

@@ -0,0 +1,105 @@
+method Study(n: nat, h: nat)
+decreases n * 201 + h
+{
+    if h != 0 {
+        // first, study for an hour, and then:
+        Study(n, h - 1);
+    } else if n == 0 {
+        // you just finished course 0 - woot woot, graduation time!
+    } else {
+        // find out how much studying is needed for the next course
+        
+        var hours := RequiredStudyTime(n - 1);
+        // get started with course n-1:
+        Study(n - 1, hours);
+    }
+}
+
+method RequiredStudyTime(c: nat) returns (hours: nat)
+ensures hours <= 200
+
+method Outer(a: nat)
+    decreases a
+{
+    if a != 0 {
+        var b := RequiredStudyTime(a - 1);
+        Inner(a, b);
+    }
+}
+
+method Inner(a: nat, b: nat)
+    requires 1 <= a
+    decreases a, b
+{
+    if b == 0 {
+        Outer(a - 1);
+    } else {
+        Inner(a, b - 1);
+    }
+}
+
+method Outer'(a: nat)
+    decreases a, 0, 0
+{
+    if a != 0 {
+        var b := RequiredStudyTime(a - 1);
+        Inner'(a - 1, b);
+    }
+}
+method Inner'(a: nat, b: nat) 
+    decreases a, b, 1
+{
+    if b == 0 {
+        Outer'(a);
+    } else {
+        Inner'(a, b - 1);
+    }
+}
+
+function F(n: nat): nat 
+decreases n, 1
+{
+    if n == 0 then 1 else n - M(F(n - 1))
+}
+
+function M(n: nat): nat 
+decreases n, 0
+{
+    if n == 0 then 0 else n - F(M(n - 1))
+}
+
+method StudyPlan(n: nat)
+    requires n <= 40
+    decreases 40 - n, 201
+{
+    if n == 40 {
+        // done
+    } else {
+        var hours := RequiredStudyTime(n);
+        Learn(n, hours);
+    }
+}
+
+method Learn(n: nat, h: nat)
+    requires n < 40
+    decreases 40 - n, h
+{
+    if h == 0 {
+        StudyPlan(n + 1);
+    } else {
+        Learn(n, h - 1);
+    }
+}
+
+function F'(x: nat, y: nat): int 
+    decreases x % 2, if x % 2 == 0 then 1000 - x else x
+{
+    if 1000 <= x then
+        x + y
+    else if x % 2 == 0 then
+        F'(x + 2, y + 1)
+    else if x < 6 then
+        F'(2 * y, y)
+    else
+        F'(x - 4, y + 3)
+}