TSTP Solution File: SET074+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET074+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 : n012.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 : Theorem 0.20s 0.48s
% 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+1 : TPTP v8.1.2. Bugfixed v5.4.0.
% 0.00/0.12 % 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 : n012.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 09:05:41 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.20/0.48 Command-line arguments: --flatten
% 0.20/0.48
% 0.20/0.48 % SZS status Theorem
% 0.20/0.48
% 0.20/0.49 % SZS output start Proof
% 0.20/0.49 Take the following subset of the input axioms:
% 0.20/0.49 fof(complement, axiom, ![X, Z]: (member(Z, complement(X)) <=> (member(Z, universal_class) & ~member(Z, X)))).
% 0.20/0.49 fof(corollary1_2, conjecture, ![Y, X2]: (member(Y, universal_class) => unordered_pair(X2, Y)!=null_class)).
% 0.20/0.49 fof(disjoint_defn, axiom, ![X2, Y2]: (disjoint(X2, Y2) <=> ![U]: ~(member(U, X2) & member(U, Y2)))).
% 0.20/0.49 fof(domain_of, axiom, ![X2, Z2]: (member(Z2, domain_of(X2)) <=> (member(Z2, universal_class) & restrict(X2, singleton(Z2), universal_class)!=null_class))).
% 0.20/0.49 fof(element_relation, axiom, subclass(element_relation, cross_product(universal_class, universal_class))).
% 0.20/0.49 fof(infinity, axiom, ?[X2]: (member(X2, universal_class) & (inductive(X2) & ![Y2]: (inductive(Y2) => subclass(X2, Y2))))).
% 0.20/0.49 fof(null_class_defn, axiom, ![X2]: ~member(X2, null_class)).
% 0.20/0.49 fof(unordered_pair_defn, axiom, ![X2, Y2, U2]: (member(U2, unordered_pair(X2, Y2)) <=> (member(U2, universal_class) & (U2=X2 | U2=Y2)))).
% 0.20/0.49
% 0.20/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.49 fresh(y, y, x1...xn) = u
% 0.20/0.49 C => fresh(s, t, x1...xn) = v
% 0.20/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.49 variables of u and v.
% 0.20/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.49 input problem has no model of domain size 1).
% 0.20/0.49
% 0.20/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.49
% 0.20/0.49 Axiom 1 (infinity_1): inductive(x3) = true2.
% 0.20/0.49 Axiom 2 (corollary1_2): unordered_pair(x, y) = null_class.
% 0.20/0.49 Axiom 3 (corollary1_2_1): member(y, universal_class) = true2.
% 0.20/0.49 Axiom 4 (infinity): member(x3, universal_class) = true2.
% 0.20/0.49 Axiom 5 (unordered_pair_defn_3): fresh8(X, X, Y, Z) = true2.
% 0.20/0.49 Axiom 6 (element_relation): subclass(element_relation, cross_product(universal_class, universal_class)) = true2.
% 0.20/0.49 Axiom 7 (unordered_pair_defn_3): fresh8(member(X, universal_class), true2, Y, X) = member(X, unordered_pair(Y, X)).
% 0.20/0.49
% 0.20/0.49 Lemma 8: member(y, universal_class) = inductive(x3).
% 0.20/0.49 Proof:
% 0.20/0.49 member(y, universal_class)
% 0.20/0.49 = { by axiom 3 (corollary1_2_1) }
% 0.20/0.49 true2
% 0.20/0.49 = { by axiom 1 (infinity_1) R->L }
% 0.20/0.49 inductive(x3)
% 0.20/0.49
% 0.20/0.49 Lemma 9: member(x3, universal_class) = member(y, universal_class).
% 0.20/0.49 Proof:
% 0.20/0.49 member(x3, universal_class)
% 0.20/0.49 = { by axiom 4 (infinity) }
% 0.20/0.49 true2
% 0.20/0.49 = { by axiom 1 (infinity_1) R->L }
% 0.20/0.49 inductive(x3)
% 0.20/0.49 = { by lemma 8 R->L }
% 0.20/0.49 member(y, universal_class)
% 0.20/0.49
% 0.20/0.49 Lemma 10: subclass(element_relation, cross_product(universal_class, universal_class)) = member(x3, universal_class).
% 0.20/0.49 Proof:
% 0.20/0.49 subclass(element_relation, cross_product(universal_class, universal_class))
% 0.20/0.49 = { by axiom 6 (element_relation) }
% 0.20/0.49 true2
% 0.20/0.49 = { by axiom 1 (infinity_1) R->L }
% 0.20/0.49 inductive(x3)
% 0.20/0.49 = { by lemma 8 R->L }
% 0.20/0.49 member(y, universal_class)
% 0.20/0.49 = { by lemma 9 R->L }
% 0.20/0.49 member(x3, universal_class)
% 0.20/0.49
% 0.20/0.49 Goal 1 (null_class_defn): member(X, null_class) = true2.
% 0.20/0.49 The goal is true when:
% 0.20/0.49 X = y
% 0.20/0.49
% 0.20/0.49 Proof:
% 0.20/0.49 member(y, null_class)
% 0.20/0.49 = { by axiom 2 (corollary1_2) R->L }
% 0.20/0.49 member(y, unordered_pair(x, y))
% 0.20/0.49 = { by axiom 7 (unordered_pair_defn_3) R->L }
% 0.20/0.49 fresh8(member(y, universal_class), true2, x, y)
% 0.20/0.49 = { by axiom 1 (infinity_1) R->L }
% 0.20/0.49 fresh8(member(y, universal_class), inductive(x3), x, y)
% 0.20/0.49 = { by lemma 8 R->L }
% 0.20/0.49 fresh8(member(y, universal_class), member(y, universal_class), x, y)
% 0.20/0.49 = { by lemma 9 R->L }
% 0.20/0.49 fresh8(member(y, universal_class), member(x3, universal_class), x, y)
% 0.20/0.49 = { by lemma 10 R->L }
% 0.20/0.49 fresh8(member(y, universal_class), subclass(element_relation, cross_product(universal_class, universal_class)), x, y)
% 0.20/0.49 = { by lemma 9 R->L }
% 0.20/0.49 fresh8(member(x3, universal_class), subclass(element_relation, cross_product(universal_class, universal_class)), x, y)
% 0.20/0.49 = { by lemma 10 R->L }
% 0.20/0.49 fresh8(subclass(element_relation, cross_product(universal_class, universal_class)), subclass(element_relation, cross_product(universal_class, universal_class)), x, y)
% 0.20/0.49 = { by axiom 5 (unordered_pair_defn_3) }
% 0.20/0.49 true2
% 0.20/0.49 = { by axiom 1 (infinity_1) R->L }
% 0.20/0.49 inductive(x3)
% 0.20/0.49 = { by lemma 8 R->L }
% 0.20/0.49 member(y, universal_class)
% 0.20/0.49 = { by lemma 9 R->L }
% 0.20/0.49 member(x3, universal_class)
% 0.20/0.49 = { by lemma 10 R->L }
% 0.20/0.49 subclass(element_relation, cross_product(universal_class, universal_class))
% 0.20/0.49 = { by lemma 10 }
% 0.20/0.49 member(x3, universal_class)
% 0.20/0.49 = { by lemma 9 }
% 0.20/0.49 member(y, universal_class)
% 0.20/0.49 = { by lemma 8 }
% 0.20/0.49 inductive(x3)
% 0.20/0.49 = { by axiom 1 (infinity_1) }
% 0.20/0.49 true2
% 0.20/0.49 % SZS output end Proof
% 0.20/0.49
% 0.20/0.49 RESULT: Theorem (the conjecture is true).
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