diff --git a/examples/example-2.spec b/examples/example-2.spec
index d93f4b9..1659662 100644
--- a/examples/example-2.spec
+++ b/examples/example-2.spec
@@ -5,37 +5,38 @@ assume: n >= 0.
 # p/1 is an auxiliary predicate
 output: q/1.
 
-# Multiplication with positive numbers preserves the order of integers
-axiom: forall N1, N2, N3 (N1 > N2 and N3 > 0 -> N1 * N3 > N2 * N3).
-
-# Induction principle instantiated for p.
-# This axiom is necessary because we use Vampire without higher-order reasoning
-axiom: p(0) and forall N (N >= 0 and p(N) -> p(N + 1)) -> forall N p(N).
-
 # Verify that q computes the floor of the square root of n
 spec: exists N (forall X (q(X) <-> X = N) and N >= 0 and N * N <= n and (N + 1) * (N + 1) > n).
 
 
 
 
+# Multiplication with positive numbers preserves the order of integers
+axiom: forall N1, N2, N3 (N1 > N2 and N3 > 0 -> N1 * N3 > N2 * N3).
+
+# Induction principle instantiated for p.
+# This axiom is necessary because we use Vampire without higher-order reasoning
+axiom: forall N1 (p(N1) and forall N2 (N2 >= N1 and not p(N2) -> not p(N2 + 1)) -> forall N2 (N2 >= N1 -> p(N2))).
+#axiom: p(0) and forall N (N >= 0 and p(N) -> p(N + 1)) -> forall N p(N).
+
 #lemma(forward): forall N N * N >= N.
 #lemma(forward): forall X (q(X) -> exists N X = N).
 #lemma(forward): forall X (q(X) <-> exists N (X = N and N >= 0 and N * N <= n and not p(N + 1))).
 #lemma(forward): exists N (q(N) <-> N >= 0 and N * N <= n and (N + 1) * (N + 1) > n).
 #lemma(forward): exists N p(N).
+#lemma(forward): forall N1, N2 (N2 > N1 and N1 >= 0 and p(N2) -> p(N1)).
 lemma(forward): forall X (p(X) <-> exists N (X = N and N >= 0 and N * N <= n)).
-lemma(forward): forall N (N >= 0 and not p(N + 1) -> (N + 1) * (N + 1) > n).
 lemma(forward): forall X (q(X) <-> exists N2 (X = N2 and N2 >= 0 and N2 * N2 <= n and (N2 + 1) * (N2 + 1) > n)).
 lemma(forward): forall N1, N2 (N1 >= 0 and N2 >= 0 and N1 < N2 -> N1 * N1 < N2 * N2).
 lemma(forward): forall N (N >= 0 and p(N + 1) -> p(N)).
 lemma(forward): not p(n + 1).
-lemma(forward): forall N1, N2 (N2 > N1 and N1 >= 0 and p(N2) -> p(N1)).
 lemma(forward): forall N1, N2 (q(N1) and N2 > N1 -> not q(N2)).
+lemma(forward): forall N (N >= 0 and not p(N + 1) -> (N + 1) * (N + 1) > n).
+
+lemma(backward): forall N1, N2 (q(N1) and q(N2) -> N1 = N2).
+axiom: forall N1, N2 (p(N1) and not p(N1 + 1) and p(N2) and not p(N2 + 1) -> N1 = N2).
+#lemma(backward): exists N (p(N) and not p(N + 1)).
+#axiom: forall N (not p(N) -> not p(N + 1)).
 
-#lemma(backward): forall N (q(N) -> p(N) and not p(N + 1)).
 lemma(backward): forall X1 (q(X1) -> p(X1) and exists X2 (exists N (X2 = N + 1 and N = X1) and not p(X2))).
-
-lemma(backward): forall N (q(N) <- p(N) and not p(N + 1)).
-
-lemma(backward): forall N (q(N) <- p(N) and not p(N + 1)).
-lemma(backward): forall X1 (q(X1) <- p(X1) and exists X2 (exists N (X2 = N + 1 and N = X1) and not p(X2))).
+#lemma(backward): forall X1 (q(X1) <- p(X1) and exists X2 (exists N (X2 = N + 1 and N = X1) and not p(X2))).