TSTP Solution File: SET065+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET065+1 : TPTP v8.1.2. Bugfixed v5.4.0.
% 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 : n017.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 : Theorem 0.18s 0.56s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SET065+1 : TPTP v8.1.2. Bugfixed v5.4.0.
% 0.06/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n017.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sat Aug 26 13:58:56 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.18/0.56 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.18/0.56
% 0.18/0.56 % SZS status Theorem
% 0.18/0.56
% 0.18/0.56 % SZS output start Proof
% 0.18/0.56 Take the following subset of the input axioms:
% 0.18/0.56 fof(class_elements_are_sets, axiom, ![X]: subclass(X, universal_class)).
% 0.18/0.56 fof(inductive_defn, axiom, ![X2]: (inductive(X2) <=> (member(null_class, X2) & subclass(image(successor_relation, X2), X2)))).
% 0.18/0.56 fof(infinity, axiom, ?[X2]: (member(X2, universal_class) & (inductive(X2) & ![Y]: (inductive(Y) => subclass(X2, Y))))).
% 0.18/0.56 fof(null_class_is_a_set, conjecture, member(null_class, universal_class)).
% 0.18/0.56 fof(subclass_defn, axiom, ![X2, Y2]: (subclass(X2, Y2) <=> ![U]: (member(U, X2) => member(U, Y2)))).
% 0.18/0.56
% 0.18/0.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.56 fresh(y, y, x1...xn) = u
% 0.18/0.56 C => fresh(s, t, x1...xn) = v
% 0.18/0.56 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.56 variables of u and v.
% 0.18/0.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.56 input problem has no model of domain size 1).
% 0.18/0.56
% 0.18/0.56 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.56
% 0.18/0.56 Axiom 1 (infinity_1): inductive(x2) = true2.
% 0.18/0.56 Axiom 2 (class_elements_are_sets): subclass(X, universal_class) = true2.
% 0.18/0.56 Axiom 3 (inductive_defn_2): fresh40(X, X, Y) = true2.
% 0.18/0.56 Axiom 4 (inductive_defn_2): fresh40(inductive(X), true2, X) = member(null_class, X).
% 0.18/0.56 Axiom 5 (subclass_defn_1): fresh22(X, X, Y, Z) = true2.
% 0.18/0.56 Axiom 6 (subclass_defn_1): fresh23(X, X, Y, Z, W) = member(W, Z).
% 0.18/0.56 Axiom 7 (subclass_defn_1): fresh23(member(X, Y), true2, Y, Z, X) = fresh22(subclass(Y, Z), true2, Z, X).
% 0.18/0.56
% 0.18/0.56 Goal 1 (null_class_is_a_set): member(null_class, universal_class) = true2.
% 0.18/0.56 Proof:
% 0.18/0.56 member(null_class, universal_class)
% 0.18/0.56 = { by axiom 6 (subclass_defn_1) R->L }
% 0.18/0.56 fresh23(true2, true2, x2, universal_class, null_class)
% 0.18/0.56 = { by axiom 3 (inductive_defn_2) R->L }
% 0.18/0.56 fresh23(fresh40(true2, true2, x2), true2, x2, universal_class, null_class)
% 0.18/0.56 = { by axiom 1 (infinity_1) R->L }
% 0.18/0.56 fresh23(fresh40(inductive(x2), true2, x2), true2, x2, universal_class, null_class)
% 0.18/0.56 = { by axiom 4 (inductive_defn_2) }
% 0.18/0.56 fresh23(member(null_class, x2), true2, x2, universal_class, null_class)
% 0.18/0.56 = { by axiom 7 (subclass_defn_1) }
% 0.18/0.56 fresh22(subclass(x2, universal_class), true2, universal_class, null_class)
% 0.18/0.56 = { by axiom 2 (class_elements_are_sets) }
% 0.18/0.56 fresh22(true2, true2, universal_class, null_class)
% 0.18/0.56 = { by axiom 5 (subclass_defn_1) }
% 0.18/0.56 true2
% 0.18/0.56 % SZS output end Proof
% 0.18/0.56
% 0.18/0.56 RESULT: Theorem (the conjecture is true).
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