TSTP Solution File: SET090+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET090+1 : TPTP v8.1.2. Bugfixed v7.3.0.
% 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 : n011.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 15:31:11 EDT 2023
% Result : Theorem 0.22s 0.58s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET090+1 : TPTP v8.1.2. Bugfixed v7.3.0.
% 0.07/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n011.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Aug 26 09:21:39 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.22/0.58 Command-line arguments: --ground-connectedness --complete-subsets
% 0.22/0.58
% 0.22/0.58 % SZS status Theorem
% 0.22/0.58
% 0.22/0.59 % SZS output start Proof
% 0.22/0.59 Take the following subset of the input axioms:
% 0.22/0.59 fof(first_second, axiom, ![X, Y]: ((member(X, universal_class) & member(Y, universal_class)) => (first(ordered_pair(X, Y))=X & second(ordered_pair(X, Y))=Y))).
% 0.22/0.59 fof(member_of_singleton, conjecture, ![U, X2]: ((member(U, universal_class) & X2=singleton(U)) => member_of(X2)=U)).
% 0.22/0.59 fof(member_singleton_singleton, axiom, ![Y2]: (member(Y2, universal_class) => singleton(member_of(singleton(Y2)))=singleton(Y2))).
% 0.22/0.59 fof(member_singleton_universal, axiom, ![Y2]: (member(Y2, universal_class) => member(member_of(singleton(Y2)), universal_class))).
% 0.22/0.59 fof(ordered_pair_defn, axiom, ![X2, Y2]: ordered_pair(X2, Y2)=unordered_pair(singleton(X2), unordered_pair(X2, singleton(Y2)))).
% 0.22/0.59
% 0.22/0.59 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.59 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.59 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.59 fresh(y, y, x1...xn) = u
% 0.22/0.59 C => fresh(s, t, x1...xn) = v
% 0.22/0.59 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.59 variables of u and v.
% 0.22/0.59 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.59 input problem has no model of domain size 1).
% 0.22/0.59
% 0.22/0.59 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.59
% 0.22/0.59 Axiom 1 (member_of_singleton): x = singleton(u).
% 0.22/0.59 Axiom 2 (member_of_singleton_1): member(u, universal_class) = true2.
% 0.22/0.59 Axiom 3 (member_singleton_singleton): fresh36(X, X, Y) = singleton(Y).
% 0.22/0.59 Axiom 4 (member_singleton_universal): fresh35(X, X, Y) = true2.
% 0.22/0.59 Axiom 5 (first_second_1): fresh57(X, X, Y, Z) = second(ordered_pair(Y, Z)).
% 0.22/0.59 Axiom 6 (first_second_1): fresh5(X, X, Y, Z) = Z.
% 0.22/0.59 Axiom 7 (member_singleton_singleton): fresh36(member(X, universal_class), true2, X) = singleton(member_of(singleton(X))).
% 0.22/0.59 Axiom 8 (member_singleton_universal): fresh35(member(X, universal_class), true2, X) = member(member_of(singleton(X)), universal_class).
% 0.22/0.59 Axiom 9 (first_second_1): fresh57(member(X, universal_class), true2, Y, X) = fresh5(member(Y, universal_class), true2, Y, X).
% 0.22/0.59 Axiom 10 (ordered_pair_defn): ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))).
% 0.22/0.59
% 0.22/0.59 Lemma 11: fresh57(member(X, universal_class), true2, u, X) = X.
% 0.22/0.59 Proof:
% 0.22/0.59 fresh57(member(X, universal_class), true2, u, X)
% 0.22/0.59 = { by axiom 9 (first_second_1) }
% 0.22/0.59 fresh5(member(u, universal_class), true2, u, X)
% 0.22/0.59 = { by axiom 2 (member_of_singleton_1) }
% 0.22/0.59 fresh5(true2, true2, u, X)
% 0.22/0.59 = { by axiom 6 (first_second_1) }
% 0.22/0.59 X
% 0.22/0.59
% 0.22/0.59 Goal 1 (member_of_singleton_2): member_of(x) = u.
% 0.22/0.59 Proof:
% 0.22/0.59 member_of(x)
% 0.22/0.59 = { by lemma 11 R->L }
% 0.22/0.59 fresh57(member(member_of(x), universal_class), true2, u, member_of(x))
% 0.22/0.59 = { by axiom 1 (member_of_singleton) }
% 0.22/0.59 fresh57(member(member_of(singleton(u)), universal_class), true2, u, member_of(x))
% 0.22/0.59 = { by axiom 8 (member_singleton_universal) R->L }
% 0.22/0.59 fresh57(fresh35(member(u, universal_class), true2, u), true2, u, member_of(x))
% 0.22/0.59 = { by axiom 2 (member_of_singleton_1) }
% 0.22/0.59 fresh57(fresh35(true2, true2, u), true2, u, member_of(x))
% 0.22/0.59 = { by axiom 4 (member_singleton_universal) }
% 0.22/0.59 fresh57(true2, true2, u, member_of(x))
% 0.22/0.59 = { by axiom 5 (first_second_1) }
% 0.22/0.59 second(ordered_pair(u, member_of(x)))
% 0.22/0.59 = { by axiom 10 (ordered_pair_defn) }
% 0.22/0.59 second(unordered_pair(singleton(u), unordered_pair(u, singleton(member_of(x)))))
% 0.22/0.59 = { by axiom 1 (member_of_singleton) R->L }
% 0.22/0.59 second(unordered_pair(x, unordered_pair(u, singleton(member_of(x)))))
% 0.22/0.59 = { by axiom 1 (member_of_singleton) }
% 0.22/0.59 second(unordered_pair(x, unordered_pair(u, singleton(member_of(singleton(u))))))
% 0.22/0.59 = { by axiom 7 (member_singleton_singleton) R->L }
% 0.22/0.59 second(unordered_pair(x, unordered_pair(u, fresh36(member(u, universal_class), true2, u))))
% 0.22/0.59 = { by axiom 2 (member_of_singleton_1) }
% 0.22/0.59 second(unordered_pair(x, unordered_pair(u, fresh36(true2, true2, u))))
% 0.22/0.59 = { by axiom 3 (member_singleton_singleton) }
% 0.22/0.59 second(unordered_pair(x, unordered_pair(u, singleton(u))))
% 0.22/0.59 = { by axiom 1 (member_of_singleton) R->L }
% 0.22/0.59 second(unordered_pair(x, unordered_pair(u, x)))
% 0.22/0.59 = { by axiom 1 (member_of_singleton) }
% 0.22/0.59 second(unordered_pair(singleton(u), unordered_pair(u, x)))
% 0.22/0.59 = { by axiom 1 (member_of_singleton) }
% 0.22/0.59 second(unordered_pair(singleton(u), unordered_pair(u, singleton(u))))
% 0.22/0.59 = { by axiom 10 (ordered_pair_defn) R->L }
% 0.22/0.59 second(ordered_pair(u, u))
% 0.22/0.59 = { by axiom 5 (first_second_1) R->L }
% 0.22/0.59 fresh57(true2, true2, u, u)
% 0.22/0.59 = { by axiom 2 (member_of_singleton_1) R->L }
% 0.22/0.59 fresh57(member(u, universal_class), true2, u, u)
% 0.22/0.59 = { by lemma 11 }
% 0.22/0.59 u
% 0.22/0.59 % SZS output end Proof
% 0.22/0.59
% 0.22/0.59 RESULT: Theorem (the conjecture is true).
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