TSTP Solution File: SET065-7 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET065-7 : 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 : n014.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 15:31:00 EDT 2023
% Result : Unsatisfiable 0.20s 0.55s
% 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 : SET065-7 : 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 : n014.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 13:33:47 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.55 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.20/0.55
% 0.20/0.55 % SZS status Unsatisfiable
% 0.20/0.55
% 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(cartesian_product3, axiom, ![X, Y, U, V]: (~member(U, X) | (~member(V, Y) | member(ordered_pair(U, V), cross_product(X, Y))))).
% 0.20/0.56 fof(corollary_2_to_cartesian_product, axiom, ![X2, Y2, U2, V2]: (~member(ordered_pair(U2, V2), cross_product(X2, Y2)) | member(V2, universal_class))).
% 0.20/0.56 fof(inductive1, axiom, ![X2]: (~inductive(X2) | member(null_class, X2))).
% 0.20/0.56 fof(omega_in_universal, axiom, member(omega, universal_class)).
% 0.20/0.56 fof(omega_is_inductive1, axiom, inductive(omega)).
% 0.20/0.56 fof(prove_null_class_is_a_set_1, negated_conjecture, ~member(null_class, universal_class)).
% 0.20/0.56
% 0.20/0.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.56 fresh(y, y, x1...xn) = u
% 0.20/0.56 C => fresh(s, t, x1...xn) = v
% 0.20/0.56 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.56 variables of u and v.
% 0.20/0.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.56 input problem has no model of domain size 1).
% 0.20/0.56
% 0.20/0.56 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.56
% 0.20/0.56 Axiom 1 (omega_in_universal): member(omega, universal_class) = true2.
% 0.20/0.56 Axiom 2 (omega_is_inductive1): inductive(omega) = true2.
% 0.20/0.56 Axiom 3 (corollary_2_to_cartesian_product): fresh60(X, X, Y) = true2.
% 0.20/0.56 Axiom 4 (inductive1): fresh41(X, X, Y) = true2.
% 0.20/0.56 Axiom 5 (inductive1): fresh41(inductive(X), true2, X) = member(null_class, X).
% 0.20/0.56 Axiom 6 (cartesian_product3): fresh71(X, X, Y, Z, W, V) = member(ordered_pair(Y, W), cross_product(Z, V)).
% 0.20/0.56 Axiom 7 (cartesian_product3): fresh70(X, X, Y, Z, W, V) = true2.
% 0.20/0.56 Axiom 8 (corollary_2_to_cartesian_product): fresh60(member(ordered_pair(X, Y), cross_product(Z, W)), true2, Y) = member(Y, universal_class).
% 0.20/0.56 Axiom 9 (cartesian_product3): fresh71(member(X, Y), true2, Z, W, X, Y) = fresh70(member(Z, W), true2, Z, W, X, Y).
% 0.20/0.56
% 0.20/0.56 Goal 1 (prove_null_class_is_a_set_1): member(null_class, universal_class) = true2.
% 0.20/0.56 Proof:
% 0.20/0.56 member(null_class, universal_class)
% 0.20/0.56 = { by axiom 8 (corollary_2_to_cartesian_product) R->L }
% 0.20/0.56 fresh60(member(ordered_pair(omega, null_class), cross_product(universal_class, omega)), true2, null_class)
% 0.20/0.56 = { by axiom 6 (cartesian_product3) R->L }
% 0.20/0.56 fresh60(fresh71(true2, true2, omega, universal_class, null_class, omega), true2, null_class)
% 0.20/0.56 = { by axiom 4 (inductive1) R->L }
% 0.20/0.56 fresh60(fresh71(fresh41(true2, true2, omega), true2, omega, universal_class, null_class, omega), true2, null_class)
% 0.20/0.56 = { by axiom 2 (omega_is_inductive1) R->L }
% 0.20/0.56 fresh60(fresh71(fresh41(inductive(omega), true2, omega), true2, omega, universal_class, null_class, omega), true2, null_class)
% 0.20/0.56 = { by axiom 5 (inductive1) }
% 0.20/0.56 fresh60(fresh71(member(null_class, omega), true2, omega, universal_class, null_class, omega), true2, null_class)
% 0.20/0.56 = { by axiom 9 (cartesian_product3) }
% 0.20/0.56 fresh60(fresh70(member(omega, universal_class), true2, omega, universal_class, null_class, omega), true2, null_class)
% 0.20/0.56 = { by axiom 1 (omega_in_universal) }
% 0.20/0.56 fresh60(fresh70(true2, true2, omega, universal_class, null_class, omega), true2, null_class)
% 0.20/0.56 = { by axiom 7 (cartesian_product3) }
% 0.20/0.56 fresh60(true2, true2, null_class)
% 0.20/0.56 = { by axiom 3 (corollary_2_to_cartesian_product) }
% 0.20/0.56 true2
% 0.20/0.56 % SZS output end Proof
% 0.20/0.56
% 0.20/0.56 RESULT: Unsatisfiable (the axioms are contradictory).
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