WIP: 075

075-is_multiply_prime.dfy

@@ -0,0 +1,60 @@
+predicate is_prime(n: int)
+{
+  n > 1 &&
+  forall k :: 2 <= k < n ==> n % k != 0
+}
+
+method is_multiply_prime(n: int) returns (can: bool)
+  requires n >= 2
+//   ensures can <==> exists k, l, m :: 2 <= k < n && 2 <= l < n && 2 <= m < n && k * l * m == n
+{
+    can := false;
+    var k := 2;
+    while k < n
+        invariant k <= n
+        // invariant can ==> exists k0, l, m :: 2 <= k0 < k && 2 <= l < n && 2 <= m < n && k0 * l * m == n
+    {
+        var can_fixed_k := false;
+
+        var l := 2;
+        while k * l <= n
+            // invariant l <= n + 1
+            invariant k * l <= n + k
+            invariant can_fixed_k ==> exists l0, m :: 2 <= l0 < l && 2 <= m < n && mul3(k, l0, m) == n
+            invariant !can_fixed_k ==> 
+                forall l0, m :: 2 <= l0 < l && 2 <= m < n ==> mul3(k, l0, m) != n
+        {
+            var can_fixed_k_l := false;
+
+            var m := 2;
+            while mul3(k, l, m) <= n
+                decreases n - mul3(k, l, m)
+                invariant m <= n
+                invariant mul3(k, l, m) <= n + k * l
+                invariant can_fixed_k_l ==> exists m0 :: 2 <= m0 < m && mul3(k, l, m0) == n
+                invariant !can_fixed_k_l ==> forall m0 :: 2 <= m0 < m ==> mul3(k, l, m0) != n
+            {
+                if mul3(k, l, m) == n {
+                    can_fixed_k_l := true;
+                }
+                assert mul3(k, l, m + 1) <= mul3(k, l, m) + k * l <= n + k * l;
+                m := m + 1;
+            }
+
+            can_fixed_k := can_fixed_k || can_fixed_k_l;
+
+            assert !can_fixed_k_l ==> forall m :: 2 <= m < n && mul3(k, l, m) <= n ==> mul3(k, l, m) != n;
+            assert k * (l + 1) <= k * l + k <= n + k;
+            assert l + 1 <= n + 1 by {}
+
+            l := l + 1;
+        }
+
+        k := k + 1;
+        can := can || can_fixed_k;
+    }
+}
+
+function mul3(a: int, b: int, c: int): int {
+    a * b * c
+}