TSTP Solution File: NUN055+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUN055+2 : TPTP v8.1.2. Released v7.3.0.
% 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 : n018.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 12:51:45 EDT 2023
% Result : Theorem 3.54s 0.85s
% Output : Proof 3.54s
% 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 : NUN055+2 : TPTP v8.1.2. Released v7.3.0.
% 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 : n018.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 : Sun Aug 27 09:24:46 EDT 2023
% 0.13/0.35 % CPUTime :
% 3.54/0.85 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 3.54/0.85
% 3.54/0.85 % SZS status Theorem
% 3.54/0.85
% 3.54/0.87 % SZS output start Proof
% 3.54/0.87 Take the following subset of the input axioms:
% 3.54/0.87 fof(axiom_1, axiom, ?[Y24]: ![X19]: ((~r1(X19) & X19!=Y24) | (r1(X19) & X19=Y24))).
% 3.54/0.87 fof(axiom_1a, axiom, ![X1, X8]: ?[Y4]: (?[Y5]: (?[Y15]: (r2(X8, Y15) & r3(X1, Y15, Y5)) & Y5=Y4) & ?[Y7]: (r2(Y7, Y4) & r3(X1, X8, Y7)))).
% 3.54/0.87 fof(axiom_2, axiom, ![X11]: ?[Y21]: ![X12]: ((~r2(X11, X12) & X12!=Y21) | (r2(X11, X12) & X12=Y21))).
% 3.54/0.87 fof(axiom_3, axiom, ![X13, X14]: ?[Y22]: ![X15]: ((~r3(X13, X14, X15) & X15!=Y22) | (r3(X13, X14, X15) & X15=Y22))).
% 3.54/0.87 fof(axiom_4a, axiom, ![X4]: ?[Y9]: (?[Y16]: (r1(Y16) & r3(X4, Y16, Y9)) & Y9=X4)).
% 3.54/0.87 fof(axiom_7a, axiom, ![X7, Y10]: (![Y20]: (~r1(Y20) | Y20!=Y10) | ~r2(X7, Y10))).
% 3.54/0.87 fof(zeroplustwoeqtwo, conjecture, ?[Y1]: (?[Y2]: (?[Y4_2]: (r2(Y4_2, Y2) & ?[Y6]: (r1(Y6) & r2(Y6, Y4_2))) & ?[Y7_2]: (r1(Y7_2) & r3(Y7_2, Y2, Y1))) & ?[Y3]: (Y1=Y3 & ?[Y5_2]: (r2(Y5_2, Y3) & ?[Y8]: (r1(Y8) & r2(Y8, Y5_2)))))).
% 3.54/0.87
% 3.54/0.87 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.54/0.87 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.54/0.87 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.54/0.87 fresh(y, y, x1...xn) = u
% 3.54/0.87 C => fresh(s, t, x1...xn) = v
% 3.54/0.87 where fresh is a fresh function symbol and x1..xn are the free
% 3.54/0.87 variables of u and v.
% 3.54/0.87 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.54/0.87 input problem has no model of domain size 1).
% 3.54/0.87
% 3.54/0.87 The encoding turns the above axioms into the following unit equations and goals:
% 3.54/0.87
% 3.54/0.87 Axiom 1 (axiom_4a): y9(X) = X.
% 3.54/0.87 Axiom 2 (axiom_4a_1): r1(y16(X)) = true2.
% 3.54/0.87 Axiom 3 (axiom_1a): y5(X, Y) = y4(X, Y).
% 3.54/0.87 Axiom 4 (axiom_1_1): fresh10(X, X, Y) = y24.
% 3.54/0.87 Axiom 5 (axiom_1a_1): r2(X, y15(Y, X)) = true2.
% 3.54/0.87 Axiom 6 (axiom_1_1): fresh10(r1(X), true2, X) = X.
% 3.54/0.87 Axiom 7 (axiom_2): fresh8(X, X, Y, Z) = true2.
% 3.54/0.87 Axiom 8 (axiom_2_1): fresh5(X, X, Y, Z) = Z.
% 3.54/0.87 Axiom 9 (axiom_1a_3): r3(X, Y, y7(X, Y)) = true2.
% 3.54/0.87 Axiom 10 (axiom_4a_2): r3(X, y16(X), y9(X)) = true2.
% 3.54/0.87 Axiom 11 (axiom_2): fresh8(X, y21(Y), Y, X) = r2(Y, X).
% 3.54/0.87 Axiom 12 (axiom_3_1): fresh4(X, X, Y, Z, W) = W.
% 3.54/0.87 Axiom 13 (axiom_1a_2): r2(y7(X, Y), y4(X, Y)) = true2.
% 3.54/0.87 Axiom 14 (axiom_2_1): fresh5(r2(X, Y), true2, X, Y) = y21(X).
% 3.54/0.87 Axiom 15 (axiom_1a_4): r3(X, y15(X, Y), y5(X, Y)) = true2.
% 3.54/0.87 Axiom 16 (axiom_3_1): fresh4(r3(X, Y, Z), true2, X, Y, Z) = y22(X, Y).
% 3.54/0.87
% 3.54/0.87 Lemma 17: y16(X) = y24.
% 3.54/0.87 Proof:
% 3.54/0.87 y16(X)
% 3.54/0.87 = { by axiom 6 (axiom_1_1) R->L }
% 3.54/0.87 fresh10(r1(y16(X)), true2, y16(X))
% 3.54/0.87 = { by axiom 2 (axiom_4a_1) }
% 3.54/0.87 fresh10(true2, true2, y16(X))
% 3.54/0.87 = { by axiom 4 (axiom_1_1) }
% 3.54/0.87 y24
% 3.54/0.87
% 3.54/0.87 Lemma 18: y15(X, Y) = y21(Y).
% 3.54/0.87 Proof:
% 3.54/0.87 y15(X, Y)
% 3.54/0.87 = { by axiom 8 (axiom_2_1) R->L }
% 3.54/0.87 fresh5(true2, true2, Y, y15(X, Y))
% 3.54/0.87 = { by axiom 5 (axiom_1a_1) R->L }
% 3.54/0.87 fresh5(r2(Y, y15(X, Y)), true2, Y, y15(X, Y))
% 3.54/0.87 = { by axiom 14 (axiom_2_1) }
% 3.54/0.87 y21(Y)
% 3.54/0.87
% 3.54/0.87 Lemma 19: y7(X, Y) = y22(X, Y).
% 3.54/0.87 Proof:
% 3.54/0.87 y7(X, Y)
% 3.54/0.87 = { by axiom 12 (axiom_3_1) R->L }
% 3.54/0.87 fresh4(true2, true2, X, Y, y7(X, Y))
% 3.54/0.87 = { by axiom 9 (axiom_1a_3) R->L }
% 3.54/0.87 fresh4(r3(X, Y, y7(X, Y)), true2, X, Y, y7(X, Y))
% 3.54/0.87 = { by axiom 16 (axiom_3_1) }
% 3.54/0.87 y22(X, Y)
% 3.54/0.87
% 3.54/0.87 Lemma 20: y22(X, y21(Y)) = y4(X, Y).
% 3.54/0.87 Proof:
% 3.54/0.87 y22(X, y21(Y))
% 3.54/0.87 = { by lemma 18 R->L }
% 3.54/0.87 y22(X, y15(X, Y))
% 3.54/0.87 = { by axiom 16 (axiom_3_1) R->L }
% 3.54/0.87 fresh4(r3(X, y15(X, Y), y4(X, Y)), true2, X, y15(X, Y), y4(X, Y))
% 3.54/0.87 = { by axiom 3 (axiom_1a) R->L }
% 3.54/0.87 fresh4(r3(X, y15(X, Y), y5(X, Y)), true2, X, y15(X, Y), y4(X, Y))
% 3.54/0.87 = { by axiom 15 (axiom_1a_4) }
% 3.54/0.87 fresh4(true2, true2, X, y15(X, Y), y4(X, Y))
% 3.54/0.87 = { by axiom 12 (axiom_3_1) }
% 3.54/0.87 y4(X, Y)
% 3.54/0.87
% 3.54/0.87 Goal 1 (zeroplustwoeqtwo): tuple(r1(X), r1(Y), r1(Z), r2(W, V), r2(X, W), r2(U, T), r2(Z, U), r3(Y, V, T)) = tuple(true2, true2, true2, true2, true2, true2, true2, true2).
% 3.54/0.87 The goal is true when:
% 3.54/0.87 X = y16(Y)
% 3.54/0.87 Y = y16(X)
% 3.54/0.87 Z = y16(X)
% 3.54/0.87 W = y15(Z, y16(Y))
% 3.54/0.87 V = y15(W, y15(Z, y16(Y)))
% 3.54/0.87 U = y7(y16(X), y15(Z, y16(Y)))
% 3.54/0.87 T = y7(y16(X), y15(W, y15(Z, y16(Y))))
% 3.54/0.87
% 3.54/0.87 Proof:
% 3.54/0.87 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), r2(y15(Z, y16(Y)), y15(W, y15(Z, y16(Y)))), r2(y16(Y), y15(Z, y16(Y))), r2(y7(y16(X), y15(Z, y16(Y))), y7(y16(X), y15(W, y15(Z, y16(Y))))), r2(y16(X), y7(y16(X), y15(Z, y16(Y)))), r3(y16(X), y15(W, y15(Z, y16(Y))), y7(y16(X), y15(W, y15(Z, y16(Y))))))
% 3.54/0.87 = { by axiom 9 (axiom_1a_3) }
% 3.54/0.87 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), r2(y15(Z, y16(Y)), y15(W, y15(Z, y16(Y)))), r2(y16(Y), y15(Z, y16(Y))), r2(y7(y16(X), y15(Z, y16(Y))), y7(y16(X), y15(W, y15(Z, y16(Y))))), r2(y16(X), y7(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.87 = { by lemma 19 }
% 3.54/0.87 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), r2(y15(Z, y16(Y)), y15(W, y15(Z, y16(Y)))), r2(y16(Y), y15(Z, y16(Y))), r2(y7(y16(X), y15(Z, y16(Y))), y22(y16(X), y15(W, y15(Z, y16(Y))))), r2(y16(X), y7(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.87 = { by axiom 5 (axiom_1a_1) }
% 3.54/0.87 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, r2(y16(Y), y15(Z, y16(Y))), r2(y7(y16(X), y15(Z, y16(Y))), y22(y16(X), y15(W, y15(Z, y16(Y))))), r2(y16(X), y7(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.87 = { by lemma 18 }
% 3.54/0.87 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, r2(y16(Y), y15(Z, y16(Y))), r2(y7(y16(X), y15(Z, y16(Y))), y22(y16(X), y21(y15(Z, y16(Y))))), r2(y16(X), y7(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.87 = { by lemma 20 }
% 3.54/0.87 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, r2(y16(Y), y15(Z, y16(Y))), r2(y7(y16(X), y15(Z, y16(Y))), y4(y16(X), y15(Z, y16(Y)))), r2(y16(X), y7(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.88 = { by axiom 13 (axiom_1a_2) }
% 3.54/0.88 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, r2(y16(Y), y15(Z, y16(Y))), true2, r2(y16(X), y7(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.88 = { by lemma 19 }
% 3.54/0.88 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, r2(y16(Y), y15(Z, y16(Y))), true2, r2(y16(X), y22(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.88 = { by axiom 5 (axiom_1a_1) }
% 3.54/0.88 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y22(y16(X), y15(Z, y16(Y)))), true2)
% 3.54/0.88 = { by lemma 18 }
% 3.54/0.88 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y22(y16(X), y21(y16(Y)))), true2)
% 3.54/0.88 = { by lemma 20 }
% 3.54/0.88 tuple(r1(y16(Y)), r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y4(y16(X), y16(Y))), true2)
% 3.54/0.88 = { by axiom 2 (axiom_4a_1) }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y4(y16(X), y16(Y))), true2)
% 3.54/0.88 = { by lemma 17 }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y4(y16(X), y24)), true2)
% 3.54/0.88 = { by axiom 8 (axiom_2_1) R->L }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), fresh5(true2, true2, y7(y16(X), y24), y4(y16(X), y24))), true2)
% 3.54/0.88 = { by axiom 13 (axiom_1a_2) R->L }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), fresh5(r2(y7(y16(X), y24), y4(y16(X), y24)), true2, y7(y16(X), y24), y4(y16(X), y24))), true2)
% 3.54/0.88 = { by axiom 14 (axiom_2_1) }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y21(y7(y16(X), y24))), true2)
% 3.54/0.88 = { by lemma 19 }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y21(y22(y16(X), y24))), true2)
% 3.54/0.88 = { by lemma 17 R->L }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y21(y22(y16(X), y16(y16(X))))), true2)
% 3.54/0.88 = { by axiom 16 (axiom_3_1) R->L }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y21(fresh4(r3(y16(X), y16(y16(X)), y16(X)), true2, y16(X), y16(y16(X)), y16(X)))), true2)
% 3.54/0.88 = { by axiom 1 (axiom_4a) R->L }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y21(fresh4(r3(y16(X), y16(y16(X)), y9(y16(X))), true2, y16(X), y16(y16(X)), y16(X)))), true2)
% 3.54/0.88 = { by axiom 10 (axiom_4a_2) }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y21(fresh4(true2, true2, y16(X), y16(y16(X)), y16(X)))), true2)
% 3.54/0.88 = { by axiom 12 (axiom_3_1) }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, r2(y16(X), y21(y16(X))), true2)
% 3.54/0.88 = { by axiom 11 (axiom_2) R->L }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, fresh8(y21(y16(X)), y21(y16(X)), y16(X), y21(y16(X))), true2)
% 3.54/0.88 = { by axiom 7 (axiom_2) }
% 3.54/0.88 tuple(true2, r1(y16(X)), r1(y16(X)), true2, true2, true2, true2, true2)
% 3.54/0.88 = { by axiom 2 (axiom_4a_1) }
% 3.54/0.88 tuple(true2, true2, r1(y16(X)), true2, true2, true2, true2, true2)
% 3.54/0.88 = { by axiom 2 (axiom_4a_1) }
% 3.54/0.88 tuple(true2, true2, true2, true2, true2, true2, true2, true2)
% 3.54/0.88 % SZS output end Proof
% 3.54/0.88
% 3.54/0.88 RESULT: Theorem (the conjecture is true).
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