TSTP Solution File: SEU180+2 by Twee---2.4.2
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- Process Solution
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% File : Twee---2.4.2
% Problem : SEU180+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 : n021.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 17:51:25 EDT 2023
% Result : Theorem 6.26s 1.20s
% Output : Proof 6.26s
% 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 : SEU180+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
% 0.16/0.34 % Computer : n021.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Wed Aug 23 16:56:39 EDT 2023
% 0.16/0.34 % CPUTime :
% 6.26/1.20 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 6.26/1.20
% 6.26/1.20 % SZS status Theorem
% 6.26/1.20
% 6.26/1.21 % SZS output start Proof
% 6.26/1.21 Take the following subset of the input axioms:
% 6.26/1.21 fof(commutativity_k2_xboole_0, axiom, ![A, B]: set_union2(A, B)=set_union2(B, A)).
% 6.26/1.21 fof(d3_tarski, axiom, ![A2, B2]: (subset(A2, B2) <=> ![C]: (in(C, A2) => in(C, B2)))).
% 6.26/1.21 fof(d6_relat_1, axiom, ![A2_2]: (relation(A2_2) => relation_field(A2_2)=set_union2(relation_dom(A2_2), relation_rng(A2_2)))).
% 6.26/1.21 fof(t20_relat_1, lemma, ![B2, C2, A2_2]: (relation(C2) => (in(ordered_pair(A2_2, B2), C2) => (in(A2_2, relation_dom(C2)) & in(B2, relation_rng(C2)))))).
% 6.26/1.21 fof(t30_relat_1, conjecture, ![A3, B2, C2]: (relation(C2) => (in(ordered_pair(A3, B2), C2) => (in(A3, relation_field(C2)) & in(B2, relation_field(C2)))))).
% 6.26/1.21 fof(t7_xboole_1, lemma, ![A3, B2]: subset(A3, set_union2(A3, B2))).
% 6.26/1.21
% 6.26/1.21 Now clausify the problem and encode Horn clauses using encoding 3 of
% 6.26/1.21 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 6.26/1.21 We repeatedly replace C & s=t => u=v by the two clauses:
% 6.26/1.21 fresh(y, y, x1...xn) = u
% 6.26/1.21 C => fresh(s, t, x1...xn) = v
% 6.26/1.21 where fresh is a fresh function symbol and x1..xn are the free
% 6.26/1.21 variables of u and v.
% 6.26/1.21 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 6.26/1.21 input problem has no model of domain size 1).
% 6.26/1.21
% 6.26/1.21 The encoding turns the above axioms into the following unit equations and goals:
% 6.26/1.21
% 6.26/1.21 Axiom 1 (t30_relat_1_1): relation(c3) = true2.
% 6.26/1.21 Axiom 2 (commutativity_k2_xboole_0): set_union2(X, Y) = set_union2(Y, X).
% 6.26/1.21 Axiom 3 (d6_relat_1): fresh126(X, X, Y) = relation_field(Y).
% 6.26/1.21 Axiom 4 (d3_tarski_1): fresh157(X, X, Y, Z) = true2.
% 6.26/1.21 Axiom 5 (t20_relat_1): fresh75(X, X, Y, Z) = true2.
% 6.26/1.21 Axiom 6 (t20_relat_1_1): fresh73(X, X, Y, Z) = true2.
% 6.26/1.21 Axiom 7 (d6_relat_1): fresh126(relation(X), true2, X) = set_union2(relation_dom(X), relation_rng(X)).
% 6.26/1.21 Axiom 8 (t7_xboole_1): subset(X, set_union2(X, Y)) = true2.
% 6.26/1.21 Axiom 9 (t30_relat_1): in(ordered_pair(a, b2), c3) = true2.
% 6.26/1.21 Axiom 10 (d3_tarski_1): fresh158(X, X, Y, Z, W) = in(W, Z).
% 6.26/1.21 Axiom 11 (t20_relat_1): fresh76(X, X, Y, Z, W) = in(Y, relation_dom(W)).
% 6.26/1.21 Axiom 12 (t20_relat_1_1): fresh74(X, X, Y, Z, W) = in(Z, relation_rng(W)).
% 6.26/1.21 Axiom 13 (d3_tarski_1): fresh158(subset(X, Y), true2, X, Y, Z) = fresh157(in(Z, X), true2, Y, Z).
% 6.26/1.21 Axiom 14 (t20_relat_1): fresh76(relation(X), true2, Y, Z, X) = fresh75(in(ordered_pair(Y, Z), X), true2, Y, X).
% 6.26/1.21 Axiom 15 (t20_relat_1_1): fresh74(relation(X), true2, Y, Z, X) = fresh73(in(ordered_pair(Y, Z), X), true2, Z, X).
% 6.26/1.21
% 6.26/1.21 Lemma 16: fresh157(in(X, Y), true2, set_union2(Y, Z), X) = in(X, set_union2(Y, Z)).
% 6.26/1.21 Proof:
% 6.26/1.21 fresh157(in(X, Y), true2, set_union2(Y, Z), X)
% 6.26/1.21 = { by axiom 13 (d3_tarski_1) R->L }
% 6.26/1.21 fresh158(subset(Y, set_union2(Y, Z)), true2, Y, set_union2(Y, Z), X)
% 6.26/1.21 = { by axiom 8 (t7_xboole_1) }
% 6.26/1.21 fresh158(true2, true2, Y, set_union2(Y, Z), X)
% 6.26/1.21 = { by axiom 10 (d3_tarski_1) }
% 6.26/1.21 in(X, set_union2(Y, Z))
% 6.26/1.21
% 6.26/1.21 Goal 1 (t30_relat_1_2): tuple2(in(a, relation_field(c3)), in(b2, relation_field(c3))) = tuple2(true2, true2).
% 6.26/1.21 Proof:
% 6.26/1.21 tuple2(in(a, relation_field(c3)), in(b2, relation_field(c3)))
% 6.26/1.21 = { by axiom 3 (d6_relat_1) R->L }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), in(b2, fresh126(true2, true2, c3)))
% 6.26/1.21 = { by axiom 1 (t30_relat_1_1) R->L }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), in(b2, fresh126(relation(c3), true2, c3)))
% 6.26/1.21 = { by axiom 7 (d6_relat_1) }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), in(b2, set_union2(relation_dom(c3), relation_rng(c3))))
% 6.26/1.21 = { by axiom 2 (commutativity_k2_xboole_0) R->L }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), in(b2, set_union2(relation_rng(c3), relation_dom(c3))))
% 6.26/1.21 = { by lemma 16 R->L }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), fresh157(in(b2, relation_rng(c3)), true2, set_union2(relation_rng(c3), relation_dom(c3)), b2))
% 6.26/1.21 = { by axiom 12 (t20_relat_1_1) R->L }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), fresh157(fresh74(true2, true2, a, b2, c3), true2, set_union2(relation_rng(c3), relation_dom(c3)), b2))
% 6.26/1.21 = { by axiom 1 (t30_relat_1_1) R->L }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), fresh157(fresh74(relation(c3), true2, a, b2, c3), true2, set_union2(relation_rng(c3), relation_dom(c3)), b2))
% 6.26/1.21 = { by axiom 15 (t20_relat_1_1) }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), fresh157(fresh73(in(ordered_pair(a, b2), c3), true2, b2, c3), true2, set_union2(relation_rng(c3), relation_dom(c3)), b2))
% 6.26/1.21 = { by axiom 9 (t30_relat_1) }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), fresh157(fresh73(true2, true2, b2, c3), true2, set_union2(relation_rng(c3), relation_dom(c3)), b2))
% 6.26/1.21 = { by axiom 6 (t20_relat_1_1) }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), fresh157(true2, true2, set_union2(relation_rng(c3), relation_dom(c3)), b2))
% 6.26/1.21 = { by axiom 4 (d3_tarski_1) }
% 6.26/1.21 tuple2(in(a, relation_field(c3)), true2)
% 6.26/1.21 = { by axiom 3 (d6_relat_1) R->L }
% 6.26/1.21 tuple2(in(a, fresh126(true2, true2, c3)), true2)
% 6.26/1.21 = { by axiom 1 (t30_relat_1_1) R->L }
% 6.26/1.21 tuple2(in(a, fresh126(relation(c3), true2, c3)), true2)
% 6.26/1.21 = { by axiom 7 (d6_relat_1) }
% 6.26/1.21 tuple2(in(a, set_union2(relation_dom(c3), relation_rng(c3))), true2)
% 6.26/1.21 = { by lemma 16 R->L }
% 6.26/1.21 tuple2(fresh157(in(a, relation_dom(c3)), true2, set_union2(relation_dom(c3), relation_rng(c3)), a), true2)
% 6.26/1.21 = { by axiom 11 (t20_relat_1) R->L }
% 6.26/1.21 tuple2(fresh157(fresh76(true2, true2, a, b2, c3), true2, set_union2(relation_dom(c3), relation_rng(c3)), a), true2)
% 6.26/1.21 = { by axiom 1 (t30_relat_1_1) R->L }
% 6.26/1.21 tuple2(fresh157(fresh76(relation(c3), true2, a, b2, c3), true2, set_union2(relation_dom(c3), relation_rng(c3)), a), true2)
% 6.26/1.21 = { by axiom 14 (t20_relat_1) }
% 6.26/1.21 tuple2(fresh157(fresh75(in(ordered_pair(a, b2), c3), true2, a, c3), true2, set_union2(relation_dom(c3), relation_rng(c3)), a), true2)
% 6.26/1.21 = { by axiom 9 (t30_relat_1) }
% 6.26/1.21 tuple2(fresh157(fresh75(true2, true2, a, c3), true2, set_union2(relation_dom(c3), relation_rng(c3)), a), true2)
% 6.26/1.21 = { by axiom 5 (t20_relat_1) }
% 6.26/1.21 tuple2(fresh157(true2, true2, set_union2(relation_dom(c3), relation_rng(c3)), a), true2)
% 6.26/1.21 = { by axiom 4 (d3_tarski_1) }
% 6.26/1.21 tuple2(true2, true2)
% 6.26/1.21 % SZS output end Proof
% 6.26/1.21
% 6.26/1.21 RESULT: Theorem (the conjecture is true).
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