TSTP Solution File: SEU167+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU167+2 : TPTP v8.1.2. Released v3.3.0.
% 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 : n001.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 17:51:21 EDT 2023
% Result : Theorem 2.18s 0.82s
% Output : Proof 2.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU167+2 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n001.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 00:24:40 EDT 2023
% 0.13/0.35 % CPUTime :
% 2.18/0.82 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 2.18/0.82
% 2.18/0.82 % SZS status Theorem
% 2.18/0.82
% 2.18/0.83 % SZS output start Proof
% 2.18/0.83 Take the following subset of the input axioms:
% 2.18/0.83 fof(l32_xboole_1, lemma, ![B, A2]: (set_difference(A2, B)=empty_set <=> subset(A2, B))).
% 2.18/0.83 fof(rc1_xboole_0, axiom, ?[A]: empty(A)).
% 2.18/0.83 fof(t118_zfmisc_1, lemma, ![C, A2_2, B2]: (subset(A2_2, B2) => (subset(cartesian_product2(A2_2, C), cartesian_product2(B2, C)) & subset(cartesian_product2(C, A2_2), cartesian_product2(C, B2))))).
% 2.18/0.83 fof(t119_zfmisc_1, conjecture, ![D, A3, B2, C2]: ((subset(A3, B2) & subset(C2, D)) => subset(cartesian_product2(A3, C2), cartesian_product2(B2, D)))).
% 2.18/0.83 fof(t33_xboole_1, lemma, ![A2_2, B2, C2]: (subset(A2_2, B2) => subset(set_difference(A2_2, C2), set_difference(B2, C2)))).
% 2.18/0.83 fof(t3_boole, axiom, ![A3]: set_difference(A3, empty_set)=A3).
% 2.18/0.83 fof(t6_boole, axiom, ![A2_2]: (empty(A2_2) => A2_2=empty_set)).
% 2.18/0.83
% 2.18/0.83 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.18/0.83 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.18/0.83 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.18/0.83 fresh(y, y, x1...xn) = u
% 2.18/0.83 C => fresh(s, t, x1...xn) = v
% 2.18/0.83 where fresh is a fresh function symbol and x1..xn are the free
% 2.18/0.83 variables of u and v.
% 2.18/0.83 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.18/0.83 input problem has no model of domain size 1).
% 2.18/0.83
% 2.18/0.83 The encoding turns the above axioms into the following unit equations and goals:
% 2.18/0.83
% 2.18/0.83 Axiom 1 (rc1_xboole_0): empty(a3) = true2.
% 2.18/0.83 Axiom 2 (t3_boole): set_difference(X, empty_set) = X.
% 2.18/0.83 Axiom 3 (t119_zfmisc_1): subset(c4, d) = true2.
% 2.18/0.83 Axiom 4 (t119_zfmisc_1_1): subset(a, b) = true2.
% 2.18/0.83 Axiom 5 (t6_boole): fresh33(X, X, Y) = empty_set.
% 2.18/0.83 Axiom 6 (l32_xboole_1): fresh68(X, X, Y, Z) = true2.
% 2.18/0.83 Axiom 7 (l32_xboole_1_1): fresh67(X, X, Y, Z) = empty_set.
% 2.18/0.83 Axiom 8 (t6_boole): fresh33(empty(X), true2, X) = X.
% 2.18/0.83 Axiom 9 (t118_zfmisc_1): fresh54(X, X, Y, Z, W) = true2.
% 2.18/0.83 Axiom 10 (t118_zfmisc_1_1): fresh53(X, X, Y, Z, W) = true2.
% 2.18/0.83 Axiom 11 (t33_xboole_1): fresh47(X, X, Y, Z, W) = true2.
% 2.18/0.83 Axiom 12 (l32_xboole_1): fresh68(set_difference(X, Y), empty_set, X, Y) = subset(X, Y).
% 2.18/0.83 Axiom 13 (l32_xboole_1_1): fresh67(subset(X, Y), true2, X, Y) = set_difference(X, Y).
% 2.18/0.83 Axiom 14 (t118_zfmisc_1): fresh54(subset(X, Y), true2, X, Y, Z) = subset(cartesian_product2(X, Z), cartesian_product2(Y, Z)).
% 2.18/0.83 Axiom 15 (t118_zfmisc_1_1): fresh53(subset(X, Y), true2, X, Y, Z) = subset(cartesian_product2(Z, X), cartesian_product2(Z, Y)).
% 2.18/0.83 Axiom 16 (t33_xboole_1): fresh47(subset(X, Y), true2, X, Y, Z) = subset(set_difference(X, Z), set_difference(Y, Z)).
% 2.18/0.83
% 2.18/0.83 Lemma 17: empty_set = a3.
% 2.18/0.83 Proof:
% 2.18/0.83 empty_set
% 2.18/0.83 = { by axiom 5 (t6_boole) R->L }
% 2.18/0.83 fresh33(true2, true2, a3)
% 2.18/0.83 = { by axiom 1 (rc1_xboole_0) R->L }
% 2.18/0.83 fresh33(empty(a3), true2, a3)
% 2.18/0.83 = { by axiom 8 (t6_boole) }
% 2.18/0.83 a3
% 2.18/0.83
% 2.18/0.83 Goal 1 (t119_zfmisc_1_2): subset(cartesian_product2(a, c4), cartesian_product2(b, d)) = true2.
% 2.18/0.83 Proof:
% 2.18/0.83 subset(cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 12 (l32_xboole_1) R->L }
% 2.18/0.83 fresh68(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 2 (t3_boole) R->L }
% 2.18/0.83 fresh68(set_difference(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), empty_set), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by lemma 17 }
% 2.18/0.83 fresh68(set_difference(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 13 (l32_xboole_1_1) R->L }
% 2.18/0.83 fresh68(fresh67(subset(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by lemma 17 R->L }
% 2.18/0.83 fresh68(fresh67(subset(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), empty_set), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 7 (l32_xboole_1_1) R->L }
% 2.18/0.83 fresh68(fresh67(subset(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), fresh67(true2, true2, cartesian_product2(b, c4), cartesian_product2(b, d))), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 10 (t118_zfmisc_1_1) R->L }
% 2.18/0.83 fresh68(fresh67(subset(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), fresh67(fresh53(true2, true2, c4, d, b), true2, cartesian_product2(b, c4), cartesian_product2(b, d))), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 3 (t119_zfmisc_1) R->L }
% 2.18/0.83 fresh68(fresh67(subset(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), fresh67(fresh53(subset(c4, d), true2, c4, d, b), true2, cartesian_product2(b, c4), cartesian_product2(b, d))), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 15 (t118_zfmisc_1_1) }
% 2.18/0.83 fresh68(fresh67(subset(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), fresh67(subset(cartesian_product2(b, c4), cartesian_product2(b, d)), true2, cartesian_product2(b, c4), cartesian_product2(b, d))), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 13 (l32_xboole_1_1) }
% 2.18/0.83 fresh68(fresh67(subset(set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), set_difference(cartesian_product2(b, c4), cartesian_product2(b, d))), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 16 (t33_xboole_1) R->L }
% 2.18/0.83 fresh68(fresh67(fresh47(subset(cartesian_product2(a, c4), cartesian_product2(b, c4)), true2, cartesian_product2(a, c4), cartesian_product2(b, c4), cartesian_product2(b, d)), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 14 (t118_zfmisc_1) R->L }
% 2.18/0.83 fresh68(fresh67(fresh47(fresh54(subset(a, b), true2, a, b, c4), true2, cartesian_product2(a, c4), cartesian_product2(b, c4), cartesian_product2(b, d)), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 4 (t119_zfmisc_1_1) }
% 2.18/0.83 fresh68(fresh67(fresh47(fresh54(true2, true2, a, b, c4), true2, cartesian_product2(a, c4), cartesian_product2(b, c4), cartesian_product2(b, d)), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 9 (t118_zfmisc_1) }
% 2.18/0.83 fresh68(fresh67(fresh47(true2, true2, cartesian_product2(a, c4), cartesian_product2(b, c4), cartesian_product2(b, d)), true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 11 (t33_xboole_1) }
% 2.18/0.83 fresh68(fresh67(true2, true2, set_difference(cartesian_product2(a, c4), cartesian_product2(b, d)), a3), empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 7 (l32_xboole_1_1) }
% 2.18/0.83 fresh68(empty_set, empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by lemma 17 }
% 2.18/0.83 fresh68(a3, empty_set, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by lemma 17 }
% 2.18/0.83 fresh68(a3, a3, cartesian_product2(a, c4), cartesian_product2(b, d))
% 2.18/0.83 = { by axiom 6 (l32_xboole_1) }
% 2.18/0.83 true2
% 2.18/0.83 % SZS output end Proof
% 2.18/0.83
% 2.18/0.83 RESULT: Theorem (the conjecture is true).
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