TSTP Solution File: SEU140+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SEU140+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 17:51:10 EDT 2023
% Result : Theorem 0.19s 0.49s
% Output : Proof 0.19s
% 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 : SEU140+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 00:17:10 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.49 Command-line arguments: --ground-connectedness --complete-subsets
% 0.19/0.49
% 0.19/0.49 % SZS status Theorem
% 0.19/0.49
% 0.19/0.50 % SZS output start Proof
% 0.19/0.50 Take the following subset of the input axioms:
% 0.19/0.50 fof(commutativity_k3_xboole_0, axiom, ![A, B]: set_intersection2(A, B)=set_intersection2(B, A)).
% 0.19/0.50 fof(d7_xboole_0, axiom, ![A2, B2]: (disjoint(A2, B2) <=> set_intersection2(A2, B2)=empty_set)).
% 0.19/0.50 fof(l32_xboole_1, lemma, ![A2_2, B2]: (set_difference(A2_2, B2)=empty_set <=> subset(A2_2, B2))).
% 0.19/0.50 fof(rc1_xboole_0, axiom, ?[A3]: empty(A3)).
% 0.19/0.50 fof(symmetry_r1_xboole_0, axiom, ![A2_2, B2]: (disjoint(A2_2, B2) => disjoint(B2, A2_2))).
% 0.19/0.50 fof(t26_xboole_1, lemma, ![C, A2_2, B2]: (subset(A2_2, B2) => subset(set_intersection2(A2_2, C), set_intersection2(B2, C)))).
% 0.19/0.50 fof(t3_boole, axiom, ![A3]: set_difference(A3, empty_set)=A3).
% 0.19/0.50 fof(t63_xboole_1, conjecture, ![A3, B2, C2]: ((subset(A3, B2) & disjoint(B2, C2)) => disjoint(A3, C2))).
% 0.19/0.50 fof(t6_boole, axiom, ![A2_2]: (empty(A2_2) => A2_2=empty_set)).
% 0.19/0.50
% 0.19/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.50 fresh(y, y, x1...xn) = u
% 0.19/0.50 C => fresh(s, t, x1...xn) = v
% 0.19/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.50 variables of u and v.
% 0.19/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.50 input problem has no model of domain size 1).
% 0.19/0.50
% 0.19/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.50
% 0.19/0.50 Axiom 1 (rc1_xboole_0): empty(a3) = true2.
% 0.19/0.50 Axiom 2 (t63_xboole_1_1): disjoint(b, c) = true2.
% 0.19/0.50 Axiom 3 (commutativity_k3_xboole_0): set_intersection2(X, Y) = set_intersection2(Y, X).
% 0.19/0.50 Axiom 4 (t3_boole): set_difference(X, empty_set) = X.
% 0.19/0.50 Axiom 5 (t63_xboole_1): subset(a, b) = true2.
% 0.19/0.50 Axiom 6 (t6_boole): fresh15(X, X, Y) = empty_set.
% 0.19/0.50 Axiom 7 (d7_xboole_0): fresh32(X, X, Y, Z) = true2.
% 0.19/0.50 Axiom 8 (d7_xboole_0_1): fresh31(X, X, Y, Z) = empty_set.
% 0.19/0.50 Axiom 9 (l32_xboole_1_1): fresh26(X, X, Y, Z) = empty_set.
% 0.19/0.50 Axiom 10 (symmetry_r1_xboole_0): fresh25(X, X, Y, Z) = true2.
% 0.19/0.50 Axiom 11 (t6_boole): fresh15(empty(X), true2, X) = X.
% 0.19/0.50 Axiom 12 (t26_xboole_1): fresh20(X, X, Y, Z, W) = true2.
% 0.19/0.50 Axiom 13 (d7_xboole_0): fresh32(set_intersection2(X, Y), empty_set, X, Y) = disjoint(X, Y).
% 0.19/0.50 Axiom 14 (d7_xboole_0_1): fresh31(disjoint(X, Y), true2, X, Y) = set_intersection2(X, Y).
% 0.19/0.50 Axiom 15 (l32_xboole_1_1): fresh26(subset(X, Y), true2, X, Y) = set_difference(X, Y).
% 0.19/0.50 Axiom 16 (symmetry_r1_xboole_0): fresh25(disjoint(X, Y), true2, X, Y) = disjoint(Y, X).
% 0.19/0.50 Axiom 17 (t26_xboole_1): fresh20(subset(X, Y), true2, X, Y, Z) = subset(set_intersection2(X, Z), set_intersection2(Y, Z)).
% 0.19/0.50
% 0.19/0.50 Lemma 18: empty_set = a3.
% 0.19/0.50 Proof:
% 0.19/0.50 empty_set
% 0.19/0.50 = { by axiom 6 (t6_boole) R->L }
% 0.19/0.50 fresh15(true2, true2, a3)
% 0.19/0.50 = { by axiom 1 (rc1_xboole_0) R->L }
% 0.19/0.50 fresh15(empty(a3), true2, a3)
% 0.19/0.50 = { by axiom 11 (t6_boole) }
% 0.19/0.50 a3
% 0.19/0.50
% 0.19/0.50 Goal 1 (t63_xboole_1_2): disjoint(a, c) = true2.
% 0.19/0.50 Proof:
% 0.19/0.50 disjoint(a, c)
% 0.19/0.50 = { by axiom 16 (symmetry_r1_xboole_0) R->L }
% 0.19/0.50 fresh25(disjoint(c, a), true2, c, a)
% 0.19/0.50 = { by axiom 13 (d7_xboole_0) R->L }
% 0.19/0.50 fresh25(fresh32(set_intersection2(c, a), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 4 (t3_boole) R->L }
% 0.19/0.50 fresh25(fresh32(set_difference(set_intersection2(c, a), empty_set), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by lemma 18 }
% 0.19/0.50 fresh25(fresh32(set_difference(set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 15 (l32_xboole_1_1) R->L }
% 0.19/0.50 fresh25(fresh32(fresh26(subset(set_intersection2(c, a), a3), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 3 (commutativity_k3_xboole_0) R->L }
% 0.19/0.50 fresh25(fresh32(fresh26(subset(set_intersection2(a, c), a3), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by lemma 18 R->L }
% 0.19/0.50 fresh25(fresh32(fresh26(subset(set_intersection2(a, c), empty_set), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 8 (d7_xboole_0_1) R->L }
% 0.19/0.50 fresh25(fresh32(fresh26(subset(set_intersection2(a, c), fresh31(true2, true2, b, c)), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 2 (t63_xboole_1_1) R->L }
% 0.19/0.50 fresh25(fresh32(fresh26(subset(set_intersection2(a, c), fresh31(disjoint(b, c), true2, b, c)), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 14 (d7_xboole_0_1) }
% 0.19/0.50 fresh25(fresh32(fresh26(subset(set_intersection2(a, c), set_intersection2(b, c)), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 17 (t26_xboole_1) R->L }
% 0.19/0.50 fresh25(fresh32(fresh26(fresh20(subset(a, b), true2, a, b, c), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 5 (t63_xboole_1) }
% 0.19/0.50 fresh25(fresh32(fresh26(fresh20(true2, true2, a, b, c), true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 12 (t26_xboole_1) }
% 0.19/0.50 fresh25(fresh32(fresh26(true2, true2, set_intersection2(c, a), a3), empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 9 (l32_xboole_1_1) }
% 0.19/0.50 fresh25(fresh32(empty_set, empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by lemma 18 }
% 0.19/0.50 fresh25(fresh32(a3, empty_set, c, a), true2, c, a)
% 0.19/0.50 = { by lemma 18 }
% 0.19/0.50 fresh25(fresh32(a3, a3, c, a), true2, c, a)
% 0.19/0.50 = { by axiom 7 (d7_xboole_0) }
% 0.19/0.50 fresh25(true2, true2, c, a)
% 0.19/0.50 = { by axiom 10 (symmetry_r1_xboole_0) }
% 0.19/0.50 true2
% 0.19/0.50 % SZS output end Proof
% 0.19/0.50
% 0.19/0.50 RESULT: Theorem (the conjecture is true).
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