TSTP Solution File: GEO147+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO147+1 : TPTP v8.1.2. Released v2.4.0.
% 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 : n031.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 : Wed Aug 30 23:27:38 EDT 2023
% Result : Theorem 0.18s 0.56s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.11 % Problem : GEO147+1 : TPTP v8.1.2. Released v2.4.0.
% 0.02/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.32 % Computer : n031.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Tue Aug 29 22:20:11 EDT 2023
% 0.11/0.32 % CPUTime :
% 0.18/0.56 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.18/0.56
% 0.18/0.56 % SZS status Theorem
% 0.18/0.56
% 0.18/0.57 % SZS output start Proof
% 0.18/0.57 Take the following subset of the input axioms:
% 0.18/0.57 fof(at_on_trajectory, axiom, ![P, X]: (once(at(X, P)) <=> incident_o(P, trajectory_of(X)))).
% 0.18/0.57 fof(conjunction_at_the_same_time, axiom, ![B, A2]: (once(at_the_same_time(A2, B)) => (once(A2) & once(B)))).
% 0.18/0.57 fof(connect_defn, axiom, ![Y, P2, X2]: (connect(X2, Y, P2) <=> once(at_the_same_time(at(X2, P2), at(Y, P2))))).
% 0.18/0.57 fof(t13, conjecture, ![P2, X2, Y2]: (connect(X2, Y2, P2) => (incident_o(P2, trajectory_of(X2)) & incident_o(P2, trajectory_of(Y2))))).
% 0.18/0.57 fof(trajectories_are_oriented_curves, axiom, ![X2]: ?[O]: trajectory_of(X2)=O).
% 0.18/0.57
% 0.18/0.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.57 fresh(y, y, x1...xn) = u
% 0.18/0.57 C => fresh(s, t, x1...xn) = v
% 0.18/0.57 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.57 variables of u and v.
% 0.18/0.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.57 input problem has no model of domain size 1).
% 0.18/0.57
% 0.18/0.57 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.57
% 0.18/0.57 Axiom 1 (trajectories_are_oriented_curves): trajectory_of(X) = o(X).
% 0.18/0.57 Axiom 2 (conjunction_at_the_same_time): fresh62(X, X, Y) = true2.
% 0.18/0.57 Axiom 3 (conjunction_at_the_same_time_1): fresh61(X, X, Y) = true2.
% 0.18/0.57 Axiom 4 (t13): connect(x, y, p) = true2.
% 0.18/0.57 Axiom 5 (at_on_trajectory_1): fresh74(X, X, Y, Z) = true2.
% 0.18/0.57 Axiom 6 (connect_defn): fresh60(X, X, Y, Z, W) = true2.
% 0.18/0.57 Axiom 7 (conjunction_at_the_same_time): fresh62(once(at_the_same_time(X, Y)), true2, X) = once(X).
% 0.18/0.57 Axiom 8 (conjunction_at_the_same_time_1): fresh61(once(at_the_same_time(X, Y)), true2, Y) = once(Y).
% 0.18/0.57 Axiom 9 (at_on_trajectory_1): fresh74(once(at(X, Y)), true2, X, Y) = incident_o(Y, trajectory_of(X)).
% 0.18/0.57 Axiom 10 (connect_defn): fresh60(connect(X, Y, Z), true2, X, Y, Z) = once(at_the_same_time(at(X, Z), at(Y, Z))).
% 0.18/0.57
% 0.18/0.57 Lemma 11: fresh74(once(at(X, Y)), true2, X, Y) = incident_o(Y, o(X)).
% 0.18/0.57 Proof:
% 0.18/0.57 fresh74(once(at(X, Y)), true2, X, Y)
% 0.18/0.57 = { by axiom 9 (at_on_trajectory_1) }
% 0.18/0.57 incident_o(Y, trajectory_of(X))
% 0.18/0.57 = { by axiom 1 (trajectories_are_oriented_curves) }
% 0.18/0.57 incident_o(Y, o(X))
% 0.18/0.57
% 0.18/0.57 Lemma 12: once(at_the_same_time(at(x, p), at(y, p))) = true2.
% 0.18/0.57 Proof:
% 0.18/0.57 once(at_the_same_time(at(x, p), at(y, p)))
% 0.18/0.57 = { by axiom 10 (connect_defn) R->L }
% 0.18/0.57 fresh60(connect(x, y, p), true2, x, y, p)
% 0.18/0.57 = { by axiom 4 (t13) }
% 0.18/0.57 fresh60(true2, true2, x, y, p)
% 0.18/0.57 = { by axiom 6 (connect_defn) }
% 0.18/0.57 true2
% 0.18/0.57
% 0.18/0.57 Goal 1 (t13_1): tuple2(incident_o(p, trajectory_of(x)), incident_o(p, trajectory_of(y))) = tuple2(true2, true2).
% 0.18/0.57 Proof:
% 0.18/0.57 tuple2(incident_o(p, trajectory_of(x)), incident_o(p, trajectory_of(y)))
% 0.18/0.57 = { by axiom 1 (trajectories_are_oriented_curves) }
% 0.18/0.57 tuple2(incident_o(p, o(x)), incident_o(p, trajectory_of(y)))
% 0.18/0.57 = { by axiom 1 (trajectories_are_oriented_curves) }
% 0.18/0.57 tuple2(incident_o(p, o(x)), incident_o(p, o(y)))
% 0.18/0.57 = { by lemma 11 R->L }
% 0.18/0.57 tuple2(fresh74(once(at(x, p)), true2, x, p), incident_o(p, o(y)))
% 0.18/0.57 = { by axiom 7 (conjunction_at_the_same_time) R->L }
% 0.18/0.57 tuple2(fresh74(fresh62(once(at_the_same_time(at(x, p), at(y, p))), true2, at(x, p)), true2, x, p), incident_o(p, o(y)))
% 0.18/0.57 = { by lemma 12 }
% 0.18/0.57 tuple2(fresh74(fresh62(true2, true2, at(x, p)), true2, x, p), incident_o(p, o(y)))
% 0.18/0.57 = { by axiom 2 (conjunction_at_the_same_time) }
% 0.18/0.57 tuple2(fresh74(true2, true2, x, p), incident_o(p, o(y)))
% 0.18/0.57 = { by axiom 5 (at_on_trajectory_1) }
% 0.18/0.57 tuple2(true2, incident_o(p, o(y)))
% 0.18/0.57 = { by lemma 11 R->L }
% 0.18/0.57 tuple2(true2, fresh74(once(at(y, p)), true2, y, p))
% 0.18/0.57 = { by axiom 8 (conjunction_at_the_same_time_1) R->L }
% 0.18/0.57 tuple2(true2, fresh74(fresh61(once(at_the_same_time(at(x, p), at(y, p))), true2, at(y, p)), true2, y, p))
% 0.18/0.57 = { by lemma 12 }
% 0.18/0.57 tuple2(true2, fresh74(fresh61(true2, true2, at(y, p)), true2, y, p))
% 0.18/0.57 = { by axiom 3 (conjunction_at_the_same_time_1) }
% 0.18/0.57 tuple2(true2, fresh74(true2, true2, y, p))
% 0.18/0.57 = { by axiom 5 (at_on_trajectory_1) }
% 0.18/0.57 tuple2(true2, true2)
% 0.18/0.57 % SZS output end Proof
% 0.18/0.57
% 0.18/0.57 RESULT: Theorem (the conjecture is true).
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