TSTP Solution File: SET083+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET083+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 : 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 15:31:08 EDT 2023
% Result : Theorem 0.21s 0.52s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SET083+1 : TPTP v8.1.2. Bugfixed v5.4.0.
% 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 08:59:15 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.52 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.21/0.52
% 0.21/0.52 % SZS status Theorem
% 0.21/0.52
% 0.21/0.53 % SZS output start Proof
% 0.21/0.53 Take the following subset of the input axioms:
% 0.21/0.53 fof(cross_product_defn, axiom, ![X, Y, U, V]: (member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y)))).
% 0.21/0.53 fof(first_second, axiom, ![X2, Y2]: ((member(X2, universal_class) & member(Y2, universal_class)) => (first(ordered_pair(X2, Y2))=X2 & second(ordered_pair(X2, Y2))=Y2))).
% 0.21/0.53 fof(infinity, axiom, ?[X2]: (member(X2, universal_class) & (inductive(X2) & ![Y2]: (inductive(Y2) => subclass(X2, Y2))))).
% 0.21/0.53 fof(ordered_pair_defn, axiom, ![X2, Y2]: ordered_pair(X2, Y2)=unordered_pair(singleton(X2), unordered_pair(X2, singleton(Y2)))).
% 0.21/0.53 fof(singleton_identified_by_element1, conjecture, ![X2, Y2]: ((singleton(X2)=singleton(Y2) & member(X2, universal_class)) => X2=Y2)).
% 0.21/0.53
% 0.21/0.53 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.53 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.53 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.53 fresh(y, y, x1...xn) = u
% 0.21/0.53 C => fresh(s, t, x1...xn) = v
% 0.21/0.53 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.53 variables of u and v.
% 0.21/0.53 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.53 input problem has no model of domain size 1).
% 0.21/0.53
% 0.21/0.53 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.53
% 0.21/0.53 Axiom 1 (singleton_identified_by_element1): singleton(x) = singleton(y).
% 0.21/0.53 Axiom 2 (infinity): member(x3, universal_class) = true2.
% 0.21/0.53 Axiom 3 (singleton_identified_by_element1_1): member(x, universal_class) = true2.
% 0.21/0.53 Axiom 4 (cross_product_defn_2): fresh62(X, X, Y, Z) = true2.
% 0.21/0.53 Axiom 5 (first_second_1): fresh55(X, X, Y, Z) = second(ordered_pair(Y, Z)).
% 0.21/0.53 Axiom 6 (first_second_1): fresh5(X, X, Y, Z) = Z.
% 0.21/0.53 Axiom 7 (cross_product_defn): fresh64(X, X, Y, Z, W, V) = true2.
% 0.21/0.53 Axiom 8 (first_second_1): fresh55(member(X, universal_class), true2, Y, X) = fresh5(member(Y, universal_class), true2, Y, X).
% 0.21/0.53 Axiom 9 (ordered_pair_defn): ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))).
% 0.21/0.53 Axiom 10 (cross_product_defn): fresh65(X, X, Y, Z, W, V) = member(ordered_pair(Y, Z), cross_product(W, V)).
% 0.21/0.53 Axiom 11 (cross_product_defn): fresh65(member(X, Y), true2, Z, X, W, Y) = fresh64(member(Z, W), true2, Z, X, W, Y).
% 0.21/0.53 Axiom 12 (cross_product_defn_2): fresh62(member(ordered_pair(X, Y), cross_product(Z, W)), true2, Y, W) = member(Y, W).
% 0.21/0.53
% 0.21/0.53 Lemma 13: ordered_pair(X, x) = ordered_pair(X, y).
% 0.21/0.53 Proof:
% 0.21/0.53 ordered_pair(X, x)
% 0.21/0.53 = { by axiom 9 (ordered_pair_defn) }
% 0.21/0.53 unordered_pair(singleton(X), unordered_pair(X, singleton(x)))
% 0.21/0.53 = { by axiom 1 (singleton_identified_by_element1) }
% 0.21/0.53 unordered_pair(singleton(X), unordered_pair(X, singleton(y)))
% 0.21/0.53 = { by axiom 9 (ordered_pair_defn) R->L }
% 0.21/0.53 ordered_pair(X, y)
% 0.21/0.53
% 0.21/0.53 Lemma 14: fresh55(member(X, universal_class), true2, x3, X) = X.
% 0.21/0.53 Proof:
% 0.21/0.53 fresh55(member(X, universal_class), true2, x3, X)
% 0.21/0.53 = { by axiom 8 (first_second_1) }
% 0.21/0.53 fresh5(member(x3, universal_class), true2, x3, X)
% 0.21/0.53 = { by axiom 2 (infinity) }
% 0.21/0.53 fresh5(true2, true2, x3, X)
% 0.21/0.53 = { by axiom 6 (first_second_1) }
% 0.21/0.53 X
% 0.21/0.53
% 0.21/0.53 Goal 1 (singleton_identified_by_element1_2): x = y.
% 0.21/0.53 Proof:
% 0.21/0.53 x
% 0.21/0.53 = { by lemma 14 R->L }
% 0.21/0.53 fresh55(member(x, universal_class), true2, x3, x)
% 0.21/0.53 = { by axiom 3 (singleton_identified_by_element1_1) }
% 0.21/0.53 fresh55(true2, true2, x3, x)
% 0.21/0.53 = { by axiom 5 (first_second_1) }
% 0.21/0.53 second(ordered_pair(x3, x))
% 0.21/0.53 = { by lemma 13 }
% 0.21/0.53 second(ordered_pair(x3, y))
% 0.21/0.53 = { by axiom 5 (first_second_1) R->L }
% 0.21/0.53 fresh55(true2, true2, x3, y)
% 0.21/0.53 = { by axiom 4 (cross_product_defn_2) R->L }
% 0.21/0.53 fresh55(fresh62(true2, true2, y, universal_class), true2, x3, y)
% 0.21/0.53 = { by axiom 7 (cross_product_defn) R->L }
% 0.21/0.53 fresh55(fresh62(fresh64(true2, true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
% 0.21/0.53 = { by axiom 2 (infinity) R->L }
% 0.21/0.53 fresh55(fresh62(fresh64(member(x3, universal_class), true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
% 0.21/0.53 = { by axiom 11 (cross_product_defn) R->L }
% 0.21/0.53 fresh55(fresh62(fresh65(member(x, universal_class), true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
% 0.21/0.53 = { by axiom 3 (singleton_identified_by_element1_1) }
% 0.21/0.53 fresh55(fresh62(fresh65(true2, true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
% 0.21/0.53 = { by axiom 10 (cross_product_defn) }
% 0.21/0.53 fresh55(fresh62(member(ordered_pair(x3, x), cross_product(universal_class, universal_class)), true2, y, universal_class), true2, x3, y)
% 0.21/0.53 = { by lemma 13 }
% 0.21/0.53 fresh55(fresh62(member(ordered_pair(x3, y), cross_product(universal_class, universal_class)), true2, y, universal_class), true2, x3, y)
% 0.21/0.53 = { by axiom 12 (cross_product_defn_2) }
% 0.21/0.53 fresh55(member(y, universal_class), true2, x3, y)
% 0.21/0.53 = { by lemma 14 }
% 0.21/0.53 y
% 0.21/0.53 % SZS output end Proof
% 0.21/0.53
% 0.21/0.53 RESULT: Theorem (the conjecture is true).
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