TSTP Solution File: SET023-3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET023-3 : TPTP v8.1.2. Released v1.0.0.
% 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 : n025.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 15:30:40 EDT 2023
% Result : Unsatisfiable 2.85s 0.83s
% Output : Proof 2.85s
% 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 : SET023-3 : TPTP v8.1.2. Released v1.0.0.
% 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 : n025.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 : Sat Aug 26 15:06:08 EDT 2023
% 0.13/0.34 % CPUTime :
% 2.85/0.83 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 2.85/0.83
% 2.85/0.83 % SZS status Unsatisfiable
% 2.85/0.83
% 2.85/0.83 % SZS output start Proof
% 2.85/0.83 Take the following subset of the input axioms:
% 2.85/0.83 fof(an_ordered_pair_predicate, hypothesis, ordered_pair_predicate(a)).
% 2.85/0.83 fof(ordered_pair_predicate1, axiom, ![X]: (~ordered_pair_predicate(X) | little_set(f2(X)))).
% 2.85/0.83 fof(ordered_pair_predicate2, axiom, ![X2]: (~ordered_pair_predicate(X2) | little_set(f3(X2)))).
% 2.85/0.83 fof(ordered_pair_predicate3, axiom, ![X2]: (~ordered_pair_predicate(X2) | X2=ordered_pair(f2(X2), f3(X2)))).
% 2.85/0.83 fof(property_of_second, axiom, ![Y, X2]: (~little_set(X2) | (~little_set(Y) | second(ordered_pair(X2, Y))=Y))).
% 2.85/0.83 fof(prove_second_component_is_small, negated_conjecture, ~little_set(second(a))).
% 2.85/0.83
% 2.85/0.83 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.85/0.83 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.85/0.83 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.85/0.83 fresh(y, y, x1...xn) = u
% 2.85/0.83 C => fresh(s, t, x1...xn) = v
% 2.85/0.83 where fresh is a fresh function symbol and x1..xn are the free
% 2.85/0.83 variables of u and v.
% 2.85/0.83 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.85/0.83 input problem has no model of domain size 1).
% 2.85/0.83
% 2.85/0.83 The encoding turns the above axioms into the following unit equations and goals:
% 2.85/0.83
% 2.85/0.83 Axiom 1 (an_ordered_pair_predicate): ordered_pair_predicate(a) = true2.
% 2.85/0.83 Axiom 2 (ordered_pair_predicate1): fresh56(X, X, Y) = true2.
% 2.85/0.83 Axiom 3 (ordered_pair_predicate2): fresh55(X, X, Y) = true2.
% 2.85/0.83 Axiom 4 (ordered_pair_predicate3): fresh19(X, X, Y) = Y.
% 2.85/0.83 Axiom 5 (property_of_second): fresh(X, X, Y, Z) = Z.
% 2.85/0.83 Axiom 6 (ordered_pair_predicate1): fresh56(ordered_pair_predicate(X), true2, X) = little_set(f2(X)).
% 2.85/0.83 Axiom 7 (ordered_pair_predicate2): fresh55(ordered_pair_predicate(X), true2, X) = little_set(f3(X)).
% 2.85/0.83 Axiom 8 (property_of_second): fresh47(X, X, Y, Z) = second(ordered_pair(Y, Z)).
% 2.85/0.83 Axiom 9 (ordered_pair_predicate3): fresh19(ordered_pair_predicate(X), true2, X) = ordered_pair(f2(X), f3(X)).
% 2.85/0.83 Axiom 10 (property_of_second): fresh47(little_set(X), true2, Y, X) = fresh(little_set(Y), true2, Y, X).
% 2.85/0.83
% 2.85/0.83 Lemma 11: little_set(f3(a)) = true2.
% 2.85/0.83 Proof:
% 2.85/0.83 little_set(f3(a))
% 2.85/0.83 = { by axiom 7 (ordered_pair_predicate2) R->L }
% 2.85/0.83 fresh55(ordered_pair_predicate(a), true2, a)
% 2.85/0.83 = { by axiom 1 (an_ordered_pair_predicate) }
% 2.85/0.83 fresh55(true2, true2, a)
% 2.85/0.83 = { by axiom 3 (ordered_pair_predicate2) }
% 2.85/0.83 true2
% 2.85/0.83
% 2.85/0.83 Goal 1 (prove_second_component_is_small): little_set(second(a)) = true2.
% 2.85/0.83 Proof:
% 2.85/0.83 little_set(second(a))
% 2.85/0.83 = { by axiom 4 (ordered_pair_predicate3) R->L }
% 2.85/0.83 little_set(second(fresh19(true2, true2, a)))
% 2.85/0.83 = { by axiom 1 (an_ordered_pair_predicate) R->L }
% 2.85/0.83 little_set(second(fresh19(ordered_pair_predicate(a), true2, a)))
% 2.85/0.83 = { by axiom 9 (ordered_pair_predicate3) }
% 2.85/0.83 little_set(second(ordered_pair(f2(a), f3(a))))
% 2.85/0.83 = { by axiom 8 (property_of_second) R->L }
% 2.85/0.83 little_set(fresh47(true2, true2, f2(a), f3(a)))
% 2.85/0.83 = { by lemma 11 R->L }
% 2.85/0.83 little_set(fresh47(little_set(f3(a)), true2, f2(a), f3(a)))
% 2.85/0.83 = { by axiom 10 (property_of_second) }
% 2.85/0.83 little_set(fresh(little_set(f2(a)), true2, f2(a), f3(a)))
% 2.85/0.83 = { by axiom 6 (ordered_pair_predicate1) R->L }
% 2.85/0.83 little_set(fresh(fresh56(ordered_pair_predicate(a), true2, a), true2, f2(a), f3(a)))
% 2.85/0.83 = { by axiom 1 (an_ordered_pair_predicate) }
% 2.85/0.83 little_set(fresh(fresh56(true2, true2, a), true2, f2(a), f3(a)))
% 2.85/0.83 = { by axiom 2 (ordered_pair_predicate1) }
% 2.85/0.83 little_set(fresh(true2, true2, f2(a), f3(a)))
% 2.85/0.83 = { by axiom 5 (property_of_second) }
% 2.85/0.83 little_set(f3(a))
% 2.85/0.83 = { by lemma 11 }
% 2.85/0.83 true2
% 2.85/0.83 % SZS output end Proof
% 2.85/0.83
% 2.85/0.83 RESULT: Unsatisfiable (the axioms are contradictory).
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