TSTP Solution File: NUN086+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUN086+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 : n015.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 12:51:55 EDT 2023
% Result : Theorem 0.22s 0.45s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : NUN086+2 : TPTP v8.1.2. Released v7.3.0.
% 0.12/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n015.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Sun Aug 27 09:44:39 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.22/0.45 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.22/0.45
% 0.22/0.45 % SZS status Theorem
% 0.22/0.45
% 0.22/0.47 % SZS output start Proof
% 0.22/0.47 Take the following subset of the input axioms:
% 0.22/0.48 fof(axiom_1, axiom, ?[Y24]: ![X19]: ((~r1(X19) & X19!=Y24) | (r1(X19) & X19=Y24))).
% 0.22/0.48 fof(axiom_2, axiom, ![X11]: ?[Y21]: ![X12]: ((~r2(X11, X12) & X12!=Y21) | (r2(X11, X12) & X12=Y21))).
% 0.22/0.48 fof(axiom_2a, axiom, ![X2, X9]: ?[Y2]: (?[Y3]: (?[Y14]: (r2(X9, Y14) & r4(X2, Y14, Y3)) & Y3=Y2) & ?[Y6]: (r3(Y6, X2, Y2) & r4(X2, X9, Y6)))).
% 0.22/0.48 fof(axiom_3, axiom, ![X13, X14]: ?[Y22]: ![X15]: ((~r3(X13, X14, X15) & X15!=Y22) | (r3(X13, X14, X15) & X15=Y22))).
% 0.22/0.48 fof(axiom_4, axiom, ![X16, X17]: ?[Y23]: ![X18]: ((~r4(X16, X17, X18) & X18!=Y23) | (r4(X16, X17, X18) & X18=Y23))).
% 0.22/0.48 fof(axiom_4a, axiom, ![X4]: ?[Y9]: (?[Y16]: (r1(Y16) & r3(X4, Y16, Y9)) & Y9=X4)).
% 0.22/0.48 fof(axiom_5a, axiom, ![X5]: ?[Y8]: (?[Y17]: (r1(Y17) & r4(X5, Y17, Y8)) & ?[Y18]: (r1(Y18) & Y8=Y18))).
% 0.22/0.48 fof(axiom_7a, axiom, ![X7, Y10]: (![Y20]: (~r1(Y20) | Y20!=Y10) | ~r2(X7, Y10))).
% 0.22/0.48 fof(zerotimesoneeqzero, conjecture, ?[Y1]: (?[Y2_2]: (?[Y3_2]: (r1(Y3_2) & r2(Y3_2, Y2_2)) & ?[Y4]: (r1(Y4) & r4(Y4, Y2_2, Y1))) & ?[Y5]: (r1(Y5) & Y1=Y5))).
% 0.22/0.48
% 0.22/0.48 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.48 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.48 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.48 fresh(y, y, x1...xn) = u
% 0.22/0.48 C => fresh(s, t, x1...xn) = v
% 0.22/0.48 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.48 variables of u and v.
% 0.22/0.48 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.48 input problem has no model of domain size 1).
% 0.22/0.48
% 0.22/0.48 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.48
% 0.22/0.48 Axiom 1 (axiom_4a): y9(X) = X.
% 0.22/0.48 Axiom 2 (axiom_5a): y8(X) = y18(X).
% 0.22/0.48 Axiom 3 (axiom_4a_1): r1(y16(X)) = true2.
% 0.22/0.48 Axiom 4 (axiom_5a_1): r1(y17(X)) = true2.
% 0.22/0.48 Axiom 5 (axiom_5a_2): r1(y18(X)) = true2.
% 0.22/0.48 Axiom 6 (axiom_2a): y3(X, Y) = y2(X, Y).
% 0.22/0.48 Axiom 7 (axiom_1_1): fresh10(X, X, Y) = y24.
% 0.22/0.48 Axiom 8 (axiom_1): fresh9(X, X, Y) = true2.
% 0.22/0.48 Axiom 9 (axiom_1): fresh9(X, y24, X) = r1(X).
% 0.22/0.48 Axiom 10 (axiom_2a_1): r2(X, y14(Y, X)) = true2.
% 0.22/0.48 Axiom 11 (axiom_1_1): fresh10(r1(X), true2, X) = X.
% 0.22/0.48 Axiom 12 (axiom_2): fresh8(X, X, Y, Z) = true2.
% 0.22/0.48 Axiom 13 (axiom_2_1): fresh5(X, X, Y, Z) = Z.
% 0.22/0.48 Axiom 14 (axiom_4a_2): r3(X, y16(X), y9(X)) = true2.
% 0.22/0.48 Axiom 15 (axiom_2a_3): r4(X, Y, y6(X, Y)) = true2.
% 0.22/0.48 Axiom 16 (axiom_5a_3): r4(X, y17(X), y8(X)) = true2.
% 0.22/0.48 Axiom 17 (axiom_2): fresh8(X, y21(Y), Y, X) = r2(Y, X).
% 0.22/0.48 Axiom 18 (axiom_3_1): fresh4(X, X, Y, Z, W) = W.
% 0.22/0.48 Axiom 19 (axiom_4_1): fresh3(X, X, Y, Z, W) = W.
% 0.22/0.48 Axiom 20 (axiom_2_1): fresh5(r2(X, Y), true2, X, Y) = y21(X).
% 0.22/0.48 Axiom 21 (axiom_2a_2): r3(y6(X, Y), X, y2(X, Y)) = true2.
% 0.22/0.48 Axiom 22 (axiom_2a_4): r4(X, y14(X, Y), y3(X, Y)) = true2.
% 0.22/0.48 Axiom 23 (axiom_3_1): fresh4(r3(X, Y, Z), true2, X, Y, Z) = y22(X, Y).
% 0.22/0.48 Axiom 24 (axiom_4_1): fresh3(r4(X, Y, Z), true2, X, Y, Z) = y23(X, Y).
% 0.22/0.48
% 0.22/0.48 Lemma 25: r1(y24) = true2.
% 0.22/0.48 Proof:
% 0.22/0.48 r1(y24)
% 0.22/0.48 = { by axiom 9 (axiom_1) R->L }
% 0.22/0.48 fresh9(y24, y24, y24)
% 0.22/0.48 = { by axiom 8 (axiom_1) }
% 0.22/0.48 true2
% 0.22/0.48
% 0.22/0.48 Lemma 26: y16(X) = y24.
% 0.22/0.48 Proof:
% 0.22/0.48 y16(X)
% 0.22/0.48 = { by axiom 11 (axiom_1_1) R->L }
% 0.22/0.48 fresh10(r1(y16(X)), true2, y16(X))
% 0.22/0.48 = { by axiom 3 (axiom_4a_1) }
% 0.22/0.48 fresh10(true2, true2, y16(X))
% 0.22/0.48 = { by axiom 7 (axiom_1_1) }
% 0.22/0.48 y24
% 0.22/0.48
% 0.22/0.48 Goal 1 (zerotimesoneeqzero): tuple(r1(X), r1(Y), r1(Z), r2(X, W), r4(Y, W, Z)) = tuple(true2, true2, true2, true2, true2).
% 0.22/0.48 The goal is true when:
% 0.22/0.48 X = y16(X)
% 0.22/0.48 Y = y24
% 0.22/0.48 Z = y24
% 0.22/0.48 W = y21(y24)
% 0.22/0.48
% 0.22/0.48 Proof:
% 0.22/0.48 tuple(r1(y16(X)), r1(y24), r1(y24), r2(y16(X), y21(y24)), r4(y24, y21(y24), y24))
% 0.22/0.48 = { by axiom 3 (axiom_4a_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y16(X), y21(y24)), r4(y24, y21(y24), y24))
% 0.22/0.48 = { by lemma 26 }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), y24))
% 0.22/0.48 = { by axiom 7 (axiom_1_1) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), fresh10(true2, true2, y8(y24))))
% 0.22/0.48 = { by axiom 5 (axiom_5a_2) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), fresh10(r1(y18(y24)), true2, y8(y24))))
% 0.22/0.48 = { by axiom 2 (axiom_5a) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), fresh10(r1(y8(y24)), true2, y8(y24))))
% 0.22/0.48 = { by axiom 11 (axiom_1_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), y8(y24)))
% 0.22/0.48 = { by axiom 19 (axiom_4_1) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), fresh3(true2, true2, y24, y17(y24), y8(y24))))
% 0.22/0.48 = { by axiom 16 (axiom_5a_3) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), fresh3(r4(y24, y17(y24), y8(y24)), true2, y24, y17(y24), y8(y24))))
% 0.22/0.48 = { by axiom 24 (axiom_4_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), y23(y24, y17(y24))))
% 0.22/0.48 = { by axiom 11 (axiom_1_1) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), y23(y24, fresh10(r1(y17(y24)), true2, y17(y24)))))
% 0.22/0.48 = { by axiom 4 (axiom_5a_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), y23(y24, fresh10(true2, true2, y17(y24)))))
% 0.22/0.48 = { by axiom 7 (axiom_1_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y21(y24), y23(y24, y24)))
% 0.22/0.48 = { by axiom 20 (axiom_2_1) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, fresh5(r2(y24, y14(y24, y24)), true2, y24, y14(y24, y24)), y23(y24, y24)))
% 0.22/0.48 = { by axiom 10 (axiom_2a_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, fresh5(true2, true2, y24, y14(y24, y24)), y23(y24, y24)))
% 0.22/0.48 = { by axiom 13 (axiom_2_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y23(y24, y24)))
% 0.22/0.48 = { by axiom 18 (axiom_3_1) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), fresh4(true2, true2, y23(y24, y24), y16(y23(y24, y24)), y23(y24, y24))))
% 0.22/0.48 = { by axiom 14 (axiom_4a_2) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), fresh4(r3(y23(y24, y24), y16(y23(y24, y24)), y9(y23(y24, y24))), true2, y23(y24, y24), y16(y23(y24, y24)), y23(y24, y24))))
% 0.22/0.48 = { by axiom 1 (axiom_4a) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), fresh4(r3(y23(y24, y24), y16(y23(y24, y24)), y23(y24, y24)), true2, y23(y24, y24), y16(y23(y24, y24)), y23(y24, y24))))
% 0.22/0.48 = { by axiom 23 (axiom_3_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y22(y23(y24, y24), y16(y23(y24, y24)))))
% 0.22/0.48 = { by lemma 26 }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y22(y23(y24, y24), y24)))
% 0.22/0.48 = { by axiom 24 (axiom_4_1) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y22(fresh3(r4(y24, y24, y6(y24, y24)), true2, y24, y24, y6(y24, y24)), y24)))
% 0.22/0.48 = { by axiom 15 (axiom_2a_3) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y22(fresh3(true2, true2, y24, y24, y6(y24, y24)), y24)))
% 0.22/0.48 = { by axiom 19 (axiom_4_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y22(y6(y24, y24), y24)))
% 0.22/0.48 = { by axiom 23 (axiom_3_1) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), fresh4(r3(y6(y24, y24), y24, y2(y24, y24)), true2, y6(y24, y24), y24, y2(y24, y24))))
% 0.22/0.48 = { by axiom 21 (axiom_2a_2) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), fresh4(true2, true2, y6(y24, y24), y24, y2(y24, y24))))
% 0.22/0.48 = { by axiom 18 (axiom_3_1) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y2(y24, y24)))
% 0.22/0.48 = { by axiom 6 (axiom_2a) R->L }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), r4(y24, y14(y24, y24), y3(y24, y24)))
% 0.22/0.48 = { by axiom 22 (axiom_2a_4) }
% 0.22/0.48 tuple(true2, r1(y24), r1(y24), r2(y24, y21(y24)), true2)
% 0.22/0.48 = { by lemma 25 }
% 0.22/0.48 tuple(true2, r1(y24), true2, r2(y24, y21(y24)), true2)
% 0.22/0.48 = { by lemma 25 }
% 0.22/0.48 tuple(true2, true2, true2, r2(y24, y21(y24)), true2)
% 0.22/0.48 = { by axiom 17 (axiom_2) R->L }
% 0.22/0.48 tuple(true2, true2, true2, fresh8(y21(y24), y21(y24), y24, y21(y24)), true2)
% 0.22/0.48 = { by axiom 12 (axiom_2) }
% 0.22/0.48 tuple(true2, true2, true2, true2, true2)
% 0.22/0.48 % SZS output end Proof
% 0.22/0.48
% 0.22/0.48 RESULT: Theorem (the conjecture is true).
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