2020-05-19 12:57:09 +02:00
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input: n -> integer.
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output: prime/1.
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assume: n >= 1.
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axiom: forall N1, N2, N3 (N1 > N2 and N3 > 0 -> N1 * N3 > N2 * N3).
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lemma: forall N N + 0 = N.
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lemma: forall I, J, N (I * J = N and I > 0 and N > 0 -> J > 0).
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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)))).
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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)))).
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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)))).
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2020-05-22 18:34:59 +02:00
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#spec: forall X (composite(X) -> p__is_integer__(X)).
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#spec: forall N (composite(N) <-> N > 1 and N <= n and exists I, J (I > 1 and J > 1 and I * J = N)).
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spec: forall X (prime(X) -> p__is_integer__(X)).
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spec: forall N (prime(N) <-> N > 1 and N <= n and not exists I, J (I > 1 and J > 1 and I * J = N)).
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