anthem-rs/examples/example-2.lemmas

# 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 X (p(X) <-> exists N (X = N and N >= 0 and N * N <= 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 (q(N1) and N2 > N1 -> not q(N2)).
lemma(forward): forall N (N >= 0 and not p(N + 1) -> (N + 1) * (N + 1) > n).


# Added completed definition of p/1:
#      forall X1 (p(X1) <-> exists N1 (0 <= N1 and N1 <= n and X1 = N1)
#                           and exists X2 (exists N2, N3 (X2 = N2 * N3 and N2 = X1 and N3 = X1) and X2 <= n))

lemma(backward): forall N1, N2 (N1 >= 0 and N2 >= 0 and N1 * N1 <= N2 * N2 -> N1 <= N2).

lemma(backward): forall N (p(N) <-> 0 <= N and N <= n and N * N <= n).
lemma(backward): forall N (p(N) -> N * N <= n).
lemma(backward): forall N (not p(N) and N >= 0 -> N * N > n).
lemma(backward): forall N (N >= 0 -> N * N < (N + 1) * (N + 1)).
lemma(backward): forall N1, N2 (p(N1) and not p(N2) and N2 >= 0 -> N1 * N1 <= n and N2 * N2 > n).
lemma(backward): forall N1, N2 (p(N1) and not p(N2) and N2 >= 0 -> N1 * N1 < N2 * N2).
lemma(backward): forall N1, N2 (p(N1) and not p(N2) and N2 >= 0 -> N1 < N2).

lemma(backward): forall N1, N2 (p(N1) and not p(N1 + 1) and p(N2) and not p(N2 + 1) -> N1 = N2).













lemma(backward): exists N (forall X (q(X) <- X = N) and N >= 0 and N * N <= n and (N + 1) * (N + 1) > n).
lemma(backward): exists N (q(N) and N >= 0 and N * N <= n and (N + 1) * (N + 1) > n).






#axiom: forall N, M (q(N) <- p(N) and not p(M) and M > N).
#axiom: forall N1, N2 (p(N1) and not p(N1 + 1) and p(N2) and not p(N2 + 1) -> N1 = N2).




#lemma(backward): forall N1, N2 (q(N1) and q(N2) -> N1 = N2).
#lemma(backward): forall N (p(N) and N > 0 -> 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))).