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Add examples

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Patrick Lühne 2020-05-06 21:39:04 +02:00
parent 70bef152a4
commit b4339bfcb3
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examples/example-0.lp Normal file
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q(X) :- p(X, Y).

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examples/example-1.lp Normal file
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q(X + Y) :- p(X), p(Y).

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axiom: forall X (isint(X) <-> exists N X = N).
axiom: forall N1, N2, N3 (N1 > N2 and N3 > 0 -> N1 * N3 > N2 * N3).
axiom: p(0) and forall N (N >= 0 and p(N) -> p(N + 1)).
input: n -> integer.
assumption: n >= 0.
assertion: exists N (forall X (q(X) <-> X = N) and N >= 0 and N * N <= n and (N + 1) * (N + 1) > n).
lemma(forward): forall N N * N >= N.
lemma(forward): forall X (q(X) -> exists N X = N).
lemma(forward): forall X (p(X) <-> exists N2 (X = N2 and N2 >= 0 and N2 * N2 <= n)).
lemma(forward): forall X (q(X) <-> exists N2 (X = N2 and N2 >= 0 and N2 * N2 <= n and not p(N2 + 1))).
lemma(forward): forall N2 (N2 >= 0 and not p(N2 + 1) -> (N2 + 1) * (N2 + 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): exists N2 (forall X (X = N2 -> (q(X) <-> N2 >= 0 and N2 * N2 <= n and (N2 + 1) * (N2 + 1) > n))).

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p(X) :- X = 0..n, X * X <= n.
q(X) :- p(X), not p(X + 1).

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axiom: forall N1, N2, N3 (N1 > N2 and N3 > 0 -> N1 * N3 > N2 * N3).
axiom: p(0) and forall N (N >= 0 and p(N) -> p(N + 1)).
input: n -> integer.
assume: n >= 0.
assert: exists N (forall X (q(X) <-> X = N) and N >= 0 and N * N <= n and (N + 1) * (N + 1) > n).
lemma(forward): forall N N * N >= N.
lemma(forward): forall X (q(X) -> exists N X = N).
lemma(forward): forall X (p(X) <-> exists N2 (X = N2 and N2 >= 0 and N2 * N2 <= n)).
lemma(forward): forall X (q(X) <-> exists N2 (X = N2 and N2 >= 0 and N2 * N2 <= n and not p(N2 + 1))).
lemma(forward): forall N2 (N2 >= 0 and not p(N2 + 1) -> (N2 + 1) * (N2 + 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): exists N2 (forall X (X = N2 -> (q(X) <-> N2 >= 0 and N2 * N2 <= n and (N2 + 1) * (N2 + 1) > n))).
lemma(forward): exists N2 p(N2).
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 N (N >= 0 -> p(N)).
lemma(forward): forall N2, N3 (q(N2) and N3 > N2 -> not q(N3)).