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Simplify examples

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This commit is contained in:
Patrick Lühne 2020-05-28 07:06:19 +02:00
parent c3b149a473
commit bd9e0bd709
Signed by: patrick
GPG Key ID: 05F3611E97A70ABF
3 changed files with 31 additions and 26 deletions
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31
examples/example-2.spec
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@ -2,29 +2,40 @@
input: n -> integer.
assume: n >= 0.
# p/1 is an auxiliary predicate, so replace all occurrences of p/1 with its completed definition
# 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)).
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).
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 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 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): 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 N2, N3 (q(N2) and N3 > N2 -> not q(N3)).
lemma(forward): forall N1, N2 (q(N1) and N2 > N1 -> not q(N2)).
#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))).

6
examples/example-exact-cover.spec
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@ -1,10 +1,10 @@
# Auxiliary predicate to determine whether a variable is integer
axiom: forall X (is_int(X) <-> exists N X = N).
#axiom: forall X (is_int(X) <-> exists N X = N).
# Perform the proofs under the assumption that n is a nonnegative integer input constant. n stands
# for the total number of input sets
input: n -> integer.
assume: n >= 0.
#assume: n >= 0.
# s/2 is the input predicate defining the sets for which the program searches for exact covers
input: s/2.
@ -15,7 +15,7 @@ output: in/1.
# Perform the proofs under the assumption that the second parameter of s/2 (the number of the set)
# is always an integer
assume: forall X, Y (s(X, Y) -> is_int(Y)).
#assume: forall X, Y (s(X, Y) -> exists N (Y = N)).
# Only valid sets can be included in the solution
spec: forall X (in(X) -> X >= 1 and X <= n).

20
examples/example-prime.spec
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@ -1,19 +1,13 @@
input: n -> integer.
output: prime/1.
# TODO: not necessary if using the lemma below in both directions
assume: n >= 1.
axiom: forall N1, N2, N3 (N1 > N2 and N3 > 0 -> N1 * N3 > N2 * N3).
lemma: forall N N + 0 = N.
lemma: forall I, J, N (I * J = N and I > 0 and N > 0 -> J > 0).
lemma(backward): forall X1 (composite(X1) <- (exists N1, N10 (X1 = N1 and 1 <= N1 and N1 <= n and 2 <= N10 and N10 <= N1 - 1 and exists N11 (N1 = (N10 * N11) and 0 < N11)))).
lemma(backward): forall N (composite(N) -> (exists N10 (1 <= N and N <= n and 2 <= N10 and N10 <= N - 1 and exists N11 (N = (N10 * N11) and 0 < N11)))).
lemma: forall X1 (composite(X1) <-> (exists N1, N10 (X1 = N1 and 1 <= N1 and N1 <= n and 2 <= N10 and N10 <= N1 - 1 and exists N11 (N1 = (N10 * N11) and 0 < N11)))).
#spec: forall X (composite(X) -> p__is_integer__(X)).
#spec: forall N (composite(N) <-> N > 1 and N <= n and exists I, J (I > 1 and J > 1 and I * J = N)).
spec: forall X (prime(X) -> p__is_integer__(X)).
spec: forall X (prime(X) -> exists N (X = N)).
spec: forall N (prime(N) <-> N > 1 and N <= n and not exists I, J (I > 1 and J > 1 and I * J = N)).
lemma(backward): forall N (composite(N) <-> N > 1 and N <= n and exists I, J (I > 1 and J > 1 and I * J = N)).