TSTP Solution File: SET090-7 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET090-7 : TPTP v8.1.2. Bugfixed v2.1.0.
% 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 : n008.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 : Unsatisfiable 0.20s 0.58s
% Output : Proof 0.20s
% 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 : SET090-7 : TPTP v8.1.2. Bugfixed v2.1.0.
% 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 : n008.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 11:46:17 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.58 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.58
% 0.20/0.58 % SZS status Unsatisfiable
% 0.20/0.58
% 0.20/0.58 % SZS output start Proof
% 0.20/0.58 Take the following subset of the input axioms:
% 0.20/0.58 fof(member_exists1, axiom, ![Y]: (~member(Y, universal_class) | member(member_of(singleton(Y)), universal_class))).
% 0.20/0.58 fof(member_exists2, axiom, ![Y2]: (~member(Y2, universal_class) | singleton(member_of(singleton(Y2)))=singleton(Y2))).
% 0.20/0.58 fof(only_member_in_singleton, axiom, ![X, Y2]: (~member(Y2, singleton(X)) | Y2=X)).
% 0.20/0.58 fof(prove_member_of_singleton_is_unique_1, negated_conjecture, member(u, universal_class)).
% 0.20/0.58 fof(prove_member_of_singleton_is_unique_2, negated_conjecture, member_of(singleton(u))!=u).
% 0.20/0.58 fof(set_in_its_singleton, axiom, ![X2]: (~member(X2, universal_class) | member(X2, singleton(X2)))).
% 0.20/0.58
% 0.20/0.58 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.58 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.58 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.58 fresh(y, y, x1...xn) = u
% 0.20/0.58 C => fresh(s, t, x1...xn) = v
% 0.20/0.58 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.58 variables of u and v.
% 0.20/0.58 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.58 input problem has no model of domain size 1).
% 0.20/0.58
% 0.20/0.58 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.58
% 0.20/0.58 Axiom 1 (prove_member_of_singleton_is_unique_1): member(u, universal_class) = true2.
% 0.20/0.58 Axiom 2 (member_exists1): fresh47(X, X, Y) = true2.
% 0.20/0.58 Axiom 3 (member_exists2): fresh46(X, X, Y) = singleton(Y).
% 0.20/0.58 Axiom 4 (set_in_its_singleton): fresh29(X, X, Y) = true2.
% 0.20/0.58 Axiom 5 (only_member_in_singleton): fresh(X, X, Y, Z) = Z.
% 0.20/0.58 Axiom 6 (member_exists1): fresh47(member(X, universal_class), true2, X) = member(member_of(singleton(X)), universal_class).
% 0.20/0.58 Axiom 7 (member_exists2): fresh46(member(X, universal_class), true2, X) = singleton(member_of(singleton(X))).
% 0.20/0.58 Axiom 8 (set_in_its_singleton): fresh29(member(X, universal_class), true2, X) = member(X, singleton(X)).
% 0.20/0.58 Axiom 9 (only_member_in_singleton): fresh(member(X, singleton(Y)), true2, X, Y) = X.
% 0.20/0.58
% 0.20/0.58 Goal 1 (prove_member_of_singleton_is_unique_2): member_of(singleton(u)) = u.
% 0.20/0.58 Proof:
% 0.20/0.58 member_of(singleton(u))
% 0.20/0.58 = { by axiom 9 (only_member_in_singleton) R->L }
% 0.20/0.58 fresh(member(member_of(singleton(u)), singleton(u)), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 3 (member_exists2) R->L }
% 0.20/0.58 fresh(member(member_of(singleton(u)), fresh46(true2, true2, u)), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 1 (prove_member_of_singleton_is_unique_1) R->L }
% 0.20/0.58 fresh(member(member_of(singleton(u)), fresh46(member(u, universal_class), true2, u)), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 7 (member_exists2) }
% 0.20/0.58 fresh(member(member_of(singleton(u)), singleton(member_of(singleton(u)))), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 8 (set_in_its_singleton) R->L }
% 0.20/0.58 fresh(fresh29(member(member_of(singleton(u)), universal_class), true2, member_of(singleton(u))), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 6 (member_exists1) R->L }
% 0.20/0.58 fresh(fresh29(fresh47(member(u, universal_class), true2, u), true2, member_of(singleton(u))), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 1 (prove_member_of_singleton_is_unique_1) }
% 0.20/0.58 fresh(fresh29(fresh47(true2, true2, u), true2, member_of(singleton(u))), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 2 (member_exists1) }
% 0.20/0.58 fresh(fresh29(true2, true2, member_of(singleton(u))), true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 4 (set_in_its_singleton) }
% 0.20/0.58 fresh(true2, true2, member_of(singleton(u)), u)
% 0.20/0.58 = { by axiom 5 (only_member_in_singleton) }
% 0.20/0.58 u
% 0.20/0.58 % SZS output end Proof
% 0.20/0.58
% 0.20/0.58 RESULT: Unsatisfiable (the axioms are contradictory).
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