TSTP Solution File: SET954+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET954+1 : TPTP v8.1.2. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 15:33:50 EDT 2023
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SET954+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.35 % Computer : n029.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Sat Aug 26 08:50:54 EDT 2023
% 0.17/0.35 % CPUTime :
% 0.20/0.40 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.40
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 Take the following subset of the input axioms:
% 0.20/0.41 fof(l55_zfmisc_1, axiom, ![B, C, D, A2]: (in(ordered_pair(A2, B), cartesian_product2(C, D)) <=> (in(A2, C) & in(B, D)))).
% 0.20/0.41 fof(t107_zfmisc_1, conjecture, ![A, B2, C2, D2]: (in(ordered_pair(A, B2), cartesian_product2(C2, D2)) => in(ordered_pair(B2, A), cartesian_product2(D2, C2)))).
% 0.20/0.41
% 0.20/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.41 fresh(y, y, x1...xn) = u
% 0.20/0.41 C => fresh(s, t, x1...xn) = v
% 0.20/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.41 variables of u and v.
% 0.20/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.41 input problem has no model of domain size 1).
% 0.20/0.41
% 0.20/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.41
% 0.20/0.41 Axiom 1 (l55_zfmisc_1_2): fresh(X, X, Y, Z) = true2.
% 0.20/0.41 Axiom 2 (l55_zfmisc_1_1): fresh2(X, X, Y, Z) = true2.
% 0.20/0.41 Axiom 3 (l55_zfmisc_1): fresh4(X, X, Y, Z, W, V) = true2.
% 0.20/0.41 Axiom 4 (l55_zfmisc_1): fresh3(X, X, Y, Z, W, V) = in(ordered_pair(Y, Z), cartesian_product2(W, V)).
% 0.20/0.41 Axiom 5 (t107_zfmisc_1): in(ordered_pair(a, b), cartesian_product2(c, d)) = true2.
% 0.20/0.41 Axiom 6 (l55_zfmisc_1): fresh3(in(X, Y), true2, Z, X, W, Y) = fresh4(in(Z, W), true2, Z, X, W, Y).
% 0.20/0.41 Axiom 7 (l55_zfmisc_1_2): fresh(in(ordered_pair(X, Y), cartesian_product2(Z, W)), true2, Y, W) = in(Y, W).
% 0.20/0.41 Axiom 8 (l55_zfmisc_1_1): fresh2(in(ordered_pair(X, Y), cartesian_product2(Z, W)), true2, X, Z) = in(X, Z).
% 0.20/0.41
% 0.20/0.41 Lemma 9: fresh3(X, X, a, b, c, d) = true2.
% 0.20/0.41 Proof:
% 0.20/0.41 fresh3(X, X, a, b, c, d)
% 0.20/0.41 = { by axiom 4 (l55_zfmisc_1) }
% 0.20/0.41 in(ordered_pair(a, b), cartesian_product2(c, d))
% 0.20/0.41 = { by axiom 5 (t107_zfmisc_1) }
% 0.20/0.41 true2
% 0.20/0.41
% 0.20/0.41 Goal 1 (t107_zfmisc_1_1): in(ordered_pair(b, a), cartesian_product2(d, c)) = true2.
% 0.20/0.41 Proof:
% 0.20/0.41 in(ordered_pair(b, a), cartesian_product2(d, c))
% 0.20/0.41 = { by axiom 4 (l55_zfmisc_1) R->L }
% 0.20/0.41 fresh3(true2, true2, b, a, d, c)
% 0.20/0.41 = { by axiom 2 (l55_zfmisc_1_1) R->L }
% 0.20/0.41 fresh3(fresh2(true2, true2, a, c), true2, b, a, d, c)
% 0.20/0.41 = { by lemma 9 R->L }
% 0.20/0.41 fresh3(fresh2(fresh3(X, X, a, b, c, d), true2, a, c), true2, b, a, d, c)
% 0.20/0.41 = { by axiom 4 (l55_zfmisc_1) }
% 0.20/0.41 fresh3(fresh2(in(ordered_pair(a, b), cartesian_product2(c, d)), true2, a, c), true2, b, a, d, c)
% 0.20/0.41 = { by axiom 8 (l55_zfmisc_1_1) }
% 0.20/0.41 fresh3(in(a, c), true2, b, a, d, c)
% 0.20/0.41 = { by axiom 6 (l55_zfmisc_1) }
% 0.20/0.41 fresh4(in(b, d), true2, b, a, d, c)
% 0.20/0.41 = { by axiom 7 (l55_zfmisc_1_2) R->L }
% 0.20/0.41 fresh4(fresh(in(ordered_pair(a, b), cartesian_product2(c, d)), true2, b, d), true2, b, a, d, c)
% 0.20/0.41 = { by axiom 4 (l55_zfmisc_1) R->L }
% 0.20/0.41 fresh4(fresh(fresh3(Y, Y, a, b, c, d), true2, b, d), true2, b, a, d, c)
% 0.20/0.41 = { by lemma 9 }
% 0.20/0.41 fresh4(fresh(true2, true2, b, d), true2, b, a, d, c)
% 0.20/0.41 = { by axiom 1 (l55_zfmisc_1_2) }
% 0.20/0.41 fresh4(true2, true2, b, a, d, c)
% 0.20/0.41 = { by axiom 3 (l55_zfmisc_1) }
% 0.20/0.41 true2
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41
% 0.20/0.41 RESULT: Theorem (the conjecture is true).
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