worked on chapter 5

chap_5/e5_3.dfy

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+lemma ex_1a(x: int, y: int) 
+    ensures 5*x - 3 * (y + x) == 2*x - 3*y
+{
+    calc {
+        5 * x -3 * (y + x);
+    ==  // Distributivity of Multiplication
+        5 * x -3 * y - 3 * x;
+    == // Commutativity
+        5 * x - 3 * x - 3 * y;
+    == // Addition
+        2 * x - 3*y;
+    }
+}
+lemma ex_1b(x: int, y: int) 
+    ensures 2 * (x + 4*y + 7) - 10 == 2*x + 8*y + 4
+{
+    calc {
+        2 * (x + 4*y + 7) - 10;
+    == // Distributivity of Multiplication
+        2 * x + 2 * 4 * y + 2 * 7 - 10;
+    == // Multiplication of integers
+        2 * x + 8 * y + 14 - 10;
+    == // Addition of integers
+        2*x + 8*y + 4;
+    }
+}
+
+lemma ex_1c(x: int, y: int) 
+    ensures 7*x + 5 < (x + 3) * (x + 4)
+    {
+    calc {
+        7*x + 5;
+    < 
+        7*x + 12;
+    <= {assert x * x >= 0;}
+        7*x + 12 + x * x;
+    ==
+        x * x + 3 * x + 4 * x + 12;
+    ==
+        (x + 3) * (x + 4);
+    }
+}
+
+// E5.6
+
+function Ack(m: nat, n: nat): nat
+    decreases m, n
+{
+    if m == 0 then
+        n + 1
+    else if n == 0 then
+        Ack(m - 1, 1)
+    else
+        Ack(m - 1, Ack(m, n - 1))
+}
+
+lemma {:induction false} IncreasingAck(n: nat)
+    ensures Ack(1, n) == n + 2
+{
+    if n == 0 {
+        calc {
+            Ack(1, n);
+        ==
+            Ack(0, 1);
+        ==
+            2;
+        }
+    } else {
+        calc {
+            Ack(1, n);
+        == // last else branch of Ack
+            Ack(0, Ack(1, n - 1));
+        == { IncreasingAck(n - 1);}
+            Ack(0, n - 1 + 2);
+        ==
+            Ack(0, n + 1);
+        ==
+            n + 1 + 1;
+        ==
+            n + 2;
+        }
+    }
+}