TSTP Solution File: SET020+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET020+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 15:30:38 EDT 2023
% Result : Theorem 0.20s 0.53s
% 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 : SET020+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
% 0.13/0.34 % Computer : n012.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Aug 26 14:41:10 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.53 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.53
% 0.20/0.53 % SZS status Theorem
% 0.20/0.53
% 0.20/0.53 % SZS output start Proof
% 0.20/0.53 Take the following subset of the input axioms:
% 0.20/0.53 fof(first_second, axiom, ![X, Y]: ((member(X, universal_class) & member(Y, universal_class)) => (first(ordered_pair(X, Y))=X & second(ordered_pair(X, Y))=Y))).
% 0.20/0.53 fof(unique_1st_and_2nd_in_pair_of_sets1, conjecture, ![U, V, X2]: ((member(U, universal_class) & (member(V, universal_class) & X2=ordered_pair(U, V))) => (first(X2)=U & second(X2)=V))).
% 0.20/0.53
% 0.20/0.53 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.53 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.53 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.53 fresh(y, y, x1...xn) = u
% 0.20/0.53 C => fresh(s, t, x1...xn) = v
% 0.20/0.53 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.53 variables of u and v.
% 0.20/0.53 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.53 input problem has no model of domain size 1).
% 0.20/0.53
% 0.20/0.53 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.53
% 0.20/0.53 Axiom 1 (unique_1st_and_2nd_in_pair_of_sets1): x = ordered_pair(u, v).
% 0.20/0.53 Axiom 2 (unique_1st_and_2nd_in_pair_of_sets1_1): member(v, universal_class) = true2.
% 0.20/0.53 Axiom 3 (unique_1st_and_2nd_in_pair_of_sets1_2): member(u, universal_class) = true2.
% 0.20/0.53 Axiom 4 (first_second): fresh56(X, X, Y, Z) = first(ordered_pair(Y, Z)).
% 0.20/0.53 Axiom 5 (first_second_1): fresh55(X, X, Y, Z) = second(ordered_pair(Y, Z)).
% 0.20/0.53 Axiom 6 (first_second_1): fresh5(X, X, Y, Z) = Z.
% 0.20/0.53 Axiom 7 (first_second): fresh4(X, X, Y, Z) = Y.
% 0.20/0.53 Axiom 8 (first_second_1): fresh55(member(X, universal_class), true2, Y, X) = fresh5(member(Y, universal_class), true2, Y, X).
% 0.20/0.53 Axiom 9 (first_second): fresh56(member(X, universal_class), true2, Y, X) = fresh4(member(Y, universal_class), true2, Y, X).
% 0.20/0.53
% 0.20/0.53 Goal 1 (unique_1st_and_2nd_in_pair_of_sets1_3): tuple4(first(x), second(x)) = tuple4(u, v).
% 0.20/0.53 Proof:
% 0.20/0.53 tuple4(first(x), second(x))
% 0.20/0.53 = { by axiom 1 (unique_1st_and_2nd_in_pair_of_sets1) }
% 0.20/0.53 tuple4(first(x), second(ordered_pair(u, v)))
% 0.20/0.53 = { by axiom 5 (first_second_1) R->L }
% 0.20/0.53 tuple4(first(x), fresh55(true2, true2, u, v))
% 0.20/0.53 = { by axiom 2 (unique_1st_and_2nd_in_pair_of_sets1_1) R->L }
% 0.20/0.53 tuple4(first(x), fresh55(member(v, universal_class), true2, u, v))
% 0.20/0.53 = { by axiom 8 (first_second_1) }
% 0.20/0.53 tuple4(first(x), fresh5(member(u, universal_class), true2, u, v))
% 0.20/0.53 = { by axiom 3 (unique_1st_and_2nd_in_pair_of_sets1_2) }
% 0.20/0.53 tuple4(first(x), fresh5(true2, true2, u, v))
% 0.20/0.53 = { by axiom 6 (first_second_1) }
% 0.20/0.53 tuple4(first(x), v)
% 0.20/0.53 = { by axiom 1 (unique_1st_and_2nd_in_pair_of_sets1) }
% 0.20/0.53 tuple4(first(ordered_pair(u, v)), v)
% 0.20/0.53 = { by axiom 4 (first_second) R->L }
% 0.20/0.53 tuple4(fresh56(true2, true2, u, v), v)
% 0.20/0.53 = { by axiom 2 (unique_1st_and_2nd_in_pair_of_sets1_1) R->L }
% 0.20/0.53 tuple4(fresh56(member(v, universal_class), true2, u, v), v)
% 0.20/0.53 = { by axiom 9 (first_second) }
% 0.20/0.53 tuple4(fresh4(member(u, universal_class), true2, u, v), v)
% 0.20/0.53 = { by axiom 3 (unique_1st_and_2nd_in_pair_of_sets1_2) }
% 0.20/0.53 tuple4(fresh4(true2, true2, u, v), v)
% 0.20/0.53 = { by axiom 7 (first_second) }
% 0.20/0.53 tuple4(u, v)
% 0.20/0.53 % SZS output end Proof
% 0.20/0.53
% 0.20/0.54 RESULT: Theorem (the conjecture is true).
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