TSTP Solution File: SET766+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET766+4 : TPTP v8.1.2. Released v2.2.0.
% 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 : n016.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 15:33:10 EDT 2023
% Result : Theorem 0.19s 0.52s
% Output : Proof 0.19s
% 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 : SET766+4 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n016.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Sat Aug 26 11:53:56 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.52 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.52
% 0.19/0.52 % SZS status Theorem
% 0.19/0.52
% 0.19/0.52 % SZS output start Proof
% 0.19/0.52 Take the following subset of the input axioms:
% 0.19/0.52 fof(equivalence, axiom, ![R, A2]: (equivalence(R, A2) <=> (![X]: (member(X, A2) => apply(R, X, X)) & (![Y, X2]: ((member(X2, A2) & member(Y, A2)) => (apply(R, X2, Y) => apply(R, Y, X2))) & ![Z, X2, Y2]: ((member(X2, A2) & (member(Y2, A2) & member(Z, A2))) => ((apply(R, X2, Y2) & apply(R, Y2, Z)) => apply(R, X2, Z))))))).
% 0.19/0.52 fof(equivalence_class, axiom, ![E, R2, X2, A2_2]: (member(X2, equivalence_class(A2_2, E, R2)) <=> (member(X2, E) & apply(R2, A2_2, X2)))).
% 0.19/0.52 fof(thIII02, conjecture, ![A, E2, R2]: ((equivalence(R2, E2) & member(A, E2)) => member(A, equivalence_class(A, E2, R2)))).
% 0.19/0.52
% 0.19/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.52 fresh(y, y, x1...xn) = u
% 0.19/0.52 C => fresh(s, t, x1...xn) = v
% 0.19/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.52 variables of u and v.
% 0.19/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.52 input problem has no model of domain size 1).
% 0.19/0.52
% 0.19/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.52
% 0.19/0.52 Axiom 1 (thIII02_1): equivalence(r, e) = true2.
% 0.19/0.52 Axiom 2 (thIII02): member(a, e) = true2.
% 0.19/0.52 Axiom 3 (equivalence_3): fresh38(X, X, Y, Z) = true2.
% 0.19/0.52 Axiom 4 (equivalence_3): fresh39(X, X, Y, Z, W) = apply(Z, W, W).
% 0.19/0.52 Axiom 5 (equivalence_class): fresh37(X, X, Y, Z, W, V) = member(V, equivalence_class(W, Z, Y)).
% 0.19/0.52 Axiom 6 (equivalence_class): fresh36(X, X, Y, Z, W, V) = true2.
% 0.19/0.52 Axiom 7 (equivalence_3): fresh39(equivalence(X, Y), true2, Y, X, Z) = fresh38(member(Z, Y), true2, X, Z).
% 0.19/0.52 Axiom 8 (equivalence_class): fresh37(apply(X, Y, Z), true2, X, W, Y, Z) = fresh36(member(Z, W), true2, X, W, Y, Z).
% 0.19/0.52
% 0.19/0.52 Goal 1 (thIII02_2): member(a, equivalence_class(a, e, r)) = true2.
% 0.19/0.52 Proof:
% 0.19/0.52 member(a, equivalence_class(a, e, r))
% 0.19/0.52 = { by axiom 5 (equivalence_class) R->L }
% 0.19/0.52 fresh37(true2, true2, r, e, a, a)
% 0.19/0.52 = { by axiom 3 (equivalence_3) R->L }
% 0.19/0.52 fresh37(fresh38(true2, true2, r, a), true2, r, e, a, a)
% 0.19/0.52 = { by axiom 2 (thIII02) R->L }
% 0.19/0.52 fresh37(fresh38(member(a, e), true2, r, a), true2, r, e, a, a)
% 0.19/0.52 = { by axiom 7 (equivalence_3) R->L }
% 0.19/0.52 fresh37(fresh39(equivalence(r, e), true2, e, r, a), true2, r, e, a, a)
% 0.19/0.52 = { by axiom 1 (thIII02_1) }
% 0.19/0.52 fresh37(fresh39(true2, true2, e, r, a), true2, r, e, a, a)
% 0.19/0.52 = { by axiom 4 (equivalence_3) }
% 0.19/0.52 fresh37(apply(r, a, a), true2, r, e, a, a)
% 0.19/0.52 = { by axiom 8 (equivalence_class) }
% 0.19/0.52 fresh36(member(a, e), true2, r, e, a, a)
% 0.19/0.52 = { by axiom 2 (thIII02) }
% 0.19/0.52 fresh36(true2, true2, r, e, a, a)
% 0.19/0.52 = { by axiom 6 (equivalence_class) }
% 0.19/0.52 true2
% 0.19/0.52 % SZS output end Proof
% 0.19/0.52
% 0.19/0.52 RESULT: Theorem (the conjecture is true).
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