TSTP Solution File: SET074-7 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET074-7 : 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 : n020.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 15:31:04 EDT 2023
% Result : Unsatisfiable 0.20s 0.56s
% 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 : SET074-7 : 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
% 0.13/0.35 % Computer : n020.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sat Aug 26 12:04:29 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.56 Command-line arguments: --ground-connectedness --complete-subsets
% 0.20/0.56
% 0.20/0.56 % SZS status Unsatisfiable
% 0.20/0.56
% 0.20/0.56 % SZS output start Proof
% 0.20/0.56 Take the following subset of the input axioms:
% 0.20/0.56 fof(complement1, axiom, ![X, Z]: (~member(Z, complement(X)) | ~member(Z, X))).
% 0.20/0.56 fof(corollary_of_null_class_is_subclass, axiom, ![X2]: (~subclass(X2, null_class) | X2=null_class)).
% 0.20/0.56 fof(domain1, axiom, ![X2, Z2]: (restrict(X2, singleton(Z2), universal_class)!=null_class | ~member(Z2, domain_of(X2)))).
% 0.20/0.56 fof(existence_of_null_class, axiom, ![Z2]: ~member(Z2, null_class)).
% 0.20/0.56 fof(prove_corollary_to_unordered_pair_axiom2_1, negated_conjecture, member(y, universal_class)).
% 0.20/0.56 fof(prove_corollary_to_unordered_pair_axiom2_2, negated_conjecture, unordered_pair(x, y)=null_class).
% 0.20/0.56 fof(singleton_in_unordered_pair2, axiom, ![Y, X2]: subclass(singleton(Y), unordered_pair(X2, Y))).
% 0.20/0.56 fof(singleton_set, axiom, ![X2]: unordered_pair(X2, X2)=singleton(X2)).
% 0.20/0.56 fof(special_classes_lemma, axiom, ![X2, Y2]: ~member(Y2, intersection(complement(X2), X2))).
% 0.20/0.56 fof(unordered_pair2, axiom, ![X2, Y2]: (~member(X2, universal_class) | member(X2, unordered_pair(X2, Y2)))).
% 0.20/0.56
% 0.20/0.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.57 fresh(y, y, x1...xn) = u
% 0.20/0.57 C => fresh(s, t, x1...xn) = v
% 0.20/0.57 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.57 variables of u and v.
% 0.20/0.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.57 input problem has no model of domain size 1).
% 0.20/0.57
% 0.20/0.57 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.57
% 0.20/0.57 Axiom 1 (singleton_set): unordered_pair(X, X) = singleton(X).
% 0.20/0.57 Axiom 2 (prove_corollary_to_unordered_pair_axiom2_2): unordered_pair(x, y) = null_class.
% 0.20/0.57 Axiom 3 (prove_corollary_to_unordered_pair_axiom2_1): member(y, universal_class) = true2.
% 0.20/0.57 Axiom 4 (corollary_of_null_class_is_subclass): fresh62(X, X, Y) = null_class.
% 0.20/0.57 Axiom 5 (unordered_pair2): fresh12(X, X, Y, Z) = true2.
% 0.20/0.57 Axiom 6 (corollary_of_null_class_is_subclass): fresh62(subclass(X, null_class), true2, X) = X.
% 0.20/0.57 Axiom 7 (singleton_in_unordered_pair2): subclass(singleton(X), unordered_pair(Y, X)) = true2.
% 0.20/0.57 Axiom 8 (unordered_pair2): fresh12(member(X, universal_class), true2, X, Y) = member(X, unordered_pair(X, Y)).
% 0.20/0.57
% 0.20/0.57 Goal 1 (existence_of_null_class): member(X, null_class) = true2.
% 0.20/0.57 The goal is true when:
% 0.20/0.57 X = y
% 0.20/0.57
% 0.20/0.57 Proof:
% 0.20/0.57 member(y, null_class)
% 0.20/0.57 = { by axiom 4 (corollary_of_null_class_is_subclass) R->L }
% 0.20/0.57 member(y, fresh62(true2, true2, singleton(y)))
% 0.20/0.57 = { by axiom 7 (singleton_in_unordered_pair2) R->L }
% 0.20/0.57 member(y, fresh62(subclass(singleton(y), unordered_pair(x, y)), true2, singleton(y)))
% 0.20/0.57 = { by axiom 2 (prove_corollary_to_unordered_pair_axiom2_2) }
% 0.20/0.57 member(y, fresh62(subclass(singleton(y), null_class), true2, singleton(y)))
% 0.20/0.57 = { by axiom 6 (corollary_of_null_class_is_subclass) }
% 0.20/0.57 member(y, singleton(y))
% 0.20/0.57 = { by axiom 1 (singleton_set) R->L }
% 0.20/0.57 member(y, unordered_pair(y, y))
% 0.20/0.57 = { by axiom 8 (unordered_pair2) R->L }
% 0.20/0.57 fresh12(member(y, universal_class), true2, y, y)
% 0.20/0.57 = { by axiom 3 (prove_corollary_to_unordered_pair_axiom2_1) }
% 0.20/0.57 fresh12(true2, true2, y, y)
% 0.20/0.57 = { by axiom 5 (unordered_pair2) }
% 0.20/0.57 true2
% 0.20/0.57 % SZS output end Proof
% 0.20/0.57
% 0.20/0.57 RESULT: Unsatisfiable (the axioms are contradictory).
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