worked on chapter 5

chap_5/e5_1.dfy

@@ -0,0 +1,34 @@
+function More(x: int): int {
+if x <= 0 then 1 else More(x - 2) + 3
+}
+
+lemma {:induction false} Increasing(x: int)
+    decreases x
+    ensures x < More(x)
+{ // 0:
+  // x is an int, x can be any value.
+    if x <= 0 { // 1: x <= 0 Therefore More(x) == 1 => x <= 0 < 1 == More(x) => x < More(x)
+    } else { // 2: Work back to basecase, x > 0 ==> repeated x - 2 eventually enters basecase. 
+        Increasing(x-2);
+    }
+}
+
+method ExampleLemmaUse(a: int) {
+    // var b := More(a);
+    // Increasing(a);
+    // Increasing(b);
+    // var c := More(b);
+    // assert 2 <= c - a;
+
+    Increasing(a); // this call is along all control paths
+    var b := More(a);
+    Increasing(b);
+    b := More(b);
+    // Increasing(b);
+    // if a < 1000 {
+    //     Increasing(More(a)); // we learn More(a) < More(More(a))
+    //     // only here, i.e., when a < 1000
+    //     assert 2 <= b - a; // this verifies
+    // }
+    assert 2 <= b - a; // this does not verify
+}