TSTP Solution File: NUN057+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUN057+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 12:51:46 EDT 2023
% Result : Theorem 63.60s 8.68s
% Output : Proof 63.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : NUN057+2 : TPTP v8.1.2. Released v7.3.0.
% 0.00/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.37 % Computer : n018.cluster.edu
% 0.15/0.37 % Model : x86_64 x86_64
% 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37 % Memory : 8042.1875MB
% 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Sun Aug 27 09:23:30 EDT 2023
% 0.15/0.37 % CPUTime :
% 63.60/8.68 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 63.60/8.68
% 63.60/8.68 % SZS status Theorem
% 63.60/8.68
% 63.60/8.69 % SZS output start Proof
% 63.60/8.69 Take the following subset of the input axioms:
% 63.60/8.69 fof(axiom_1, axiom, ?[Y24]: ![X19]: ((~r1(X19) & X19!=Y24) | (r1(X19) & X19=Y24))).
% 63.60/8.69 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)))).
% 63.60/8.69 fof(axiom_2, axiom, ![X11]: ?[Y21]: ![X12]: ((~r2(X11, X12) & X12!=Y21) | (r2(X11, X12) & X12=Y21))).
% 63.60/8.69 fof(axiom_3, axiom, ![X13, X14]: ?[Y22]: ![X15]: ((~r3(X13, X14, X15) & X15!=Y22) | (r3(X13, X14, X15) & X15=Y22))).
% 63.60/8.69 fof(axiom_4a, axiom, ![X4]: ?[Y9]: (?[Y16]: (r1(Y16) & r3(X4, Y16, Y9)) & Y9=X4)).
% 63.60/8.69 fof(axiom_7a, axiom, ![X7, Y10]: (![Y20]: (~r1(Y20) | Y20!=Y10) | ~r2(X7, Y10))).
% 63.60/8.70 fof(twoplustwoeqfour, conjecture, ?[Y1]: (?[Y2]: (Y1=Y2 & ?[Y3]: (r2(Y3, Y2) & ?[Y4_2]: (r2(Y4_2, Y3) & ?[Y6]: (r2(Y6, Y4_2) & ?[Y8]: (r1(Y8) & r2(Y8, Y6)))))) & ?[Y5_2]: (r3(Y5_2, Y5_2, Y1) & ?[Y7_2]: (r2(Y7_2, Y5_2) & ?[Y9_2]: (r1(Y9_2) & r2(Y9_2, Y7_2)))))).
% 63.60/8.70
% 63.60/8.70 Now clausify the problem and encode Horn clauses using encoding 3 of
% 63.60/8.70 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 63.60/8.70 We repeatedly replace C & s=t => u=v by the two clauses:
% 63.60/8.70 fresh(y, y, x1...xn) = u
% 63.60/8.70 C => fresh(s, t, x1...xn) = v
% 63.60/8.70 where fresh is a fresh function symbol and x1..xn are the free
% 63.60/8.70 variables of u and v.
% 63.60/8.70 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 63.60/8.70 input problem has no model of domain size 1).
% 63.60/8.70
% 63.60/8.70 The encoding turns the above axioms into the following unit equations and goals:
% 63.60/8.70
% 63.60/8.70 Axiom 1 (axiom_4a): y9(X) = X.
% 63.60/8.70 Axiom 2 (axiom_4a_1): r1(y16(X)) = true2.
% 63.60/8.70 Axiom 3 (axiom_1a): y5(X, Y) = y4(X, Y).
% 63.60/8.70 Axiom 4 (axiom_1_1): fresh10(X, X, Y) = y24.
% 63.60/8.70 Axiom 5 (axiom_1): fresh9(X, X, Y) = true2.
% 63.60/8.70 Axiom 6 (axiom_1): fresh9(X, y24, X) = r1(X).
% 63.60/8.70 Axiom 7 (axiom_1a_1): r2(X, y15(Y, X)) = true2.
% 63.60/8.70 Axiom 8 (axiom_1_1): fresh10(r1(X), true2, X) = X.
% 63.60/8.70 Axiom 9 (axiom_2): fresh8(X, X, Y, Z) = true2.
% 63.60/8.70 Axiom 10 (axiom_2_1): fresh5(X, X, Y, Z) = Z.
% 63.60/8.70 Axiom 11 (axiom_1a_3): r3(X, Y, y7(X, Y)) = true2.
% 63.60/8.70 Axiom 12 (axiom_4a_2): r3(X, y16(X), y9(X)) = true2.
% 63.60/8.70 Axiom 13 (axiom_2): fresh8(X, y21(Y), Y, X) = r2(Y, X).
% 63.60/8.70 Axiom 14 (axiom_3_1): fresh4(X, X, Y, Z, W) = W.
% 63.60/8.70 Axiom 15 (axiom_1a_2): r2(y7(X, Y), y4(X, Y)) = true2.
% 63.60/8.70 Axiom 16 (axiom_2_1): fresh5(r2(X, Y), true2, X, Y) = y21(X).
% 63.60/8.70 Axiom 17 (axiom_1a_4): r3(X, y15(X, Y), y5(X, Y)) = true2.
% 63.60/8.70 Axiom 18 (axiom_3_1): fresh4(r3(X, Y, Z), true2, X, Y, Z) = y22(X, Y).
% 63.60/8.70
% 63.60/8.70 Lemma 19: r1(y24) = true2.
% 63.60/8.70 Proof:
% 63.60/8.70 r1(y24)
% 63.60/8.70 = { by axiom 6 (axiom_1) R->L }
% 63.60/8.70 fresh9(y24, y24, y24)
% 63.60/8.70 = { by axiom 5 (axiom_1) }
% 63.60/8.70 true2
% 63.60/8.70
% 63.60/8.70 Lemma 20: y7(X, Y) = y22(X, Y).
% 63.60/8.70 Proof:
% 63.60/8.70 y7(X, Y)
% 63.60/8.70 = { by axiom 14 (axiom_3_1) R->L }
% 63.60/8.70 fresh4(true2, true2, X, Y, y7(X, Y))
% 63.60/8.70 = { by axiom 11 (axiom_1a_3) R->L }
% 63.60/8.70 fresh4(r3(X, Y, y7(X, Y)), true2, X, Y, y7(X, Y))
% 63.60/8.70 = { by axiom 18 (axiom_3_1) }
% 63.60/8.70 y22(X, Y)
% 63.60/8.70
% 63.60/8.70 Lemma 21: r2(X, y21(X)) = true2.
% 63.60/8.70 Proof:
% 63.60/8.70 r2(X, y21(X))
% 63.60/8.70 = { by axiom 13 (axiom_2) R->L }
% 63.60/8.70 fresh8(y21(X), y21(X), X, y21(X))
% 63.60/8.70 = { by axiom 9 (axiom_2) }
% 63.60/8.70 true2
% 63.60/8.70
% 63.60/8.70 Lemma 22: y22(X, y21(Y)) = y4(X, Y).
% 63.60/8.70 Proof:
% 63.60/8.70 y22(X, y21(Y))
% 63.60/8.70 = { by axiom 16 (axiom_2_1) R->L }
% 63.60/8.70 y22(X, fresh5(r2(Y, y15(X, Y)), true2, Y, y15(X, Y)))
% 63.60/8.70 = { by axiom 7 (axiom_1a_1) }
% 63.60/8.70 y22(X, fresh5(true2, true2, Y, y15(X, Y)))
% 63.60/8.70 = { by axiom 10 (axiom_2_1) }
% 63.60/8.70 y22(X, y15(X, Y))
% 63.60/8.70 = { by axiom 18 (axiom_3_1) R->L }
% 63.60/8.70 fresh4(r3(X, y15(X, Y), y4(X, Y)), true2, X, y15(X, Y), y4(X, Y))
% 63.60/8.70 = { by axiom 3 (axiom_1a) R->L }
% 63.60/8.70 fresh4(r3(X, y15(X, Y), y5(X, Y)), true2, X, y15(X, Y), y4(X, Y))
% 63.60/8.70 = { by axiom 17 (axiom_1a_4) }
% 63.60/8.70 fresh4(true2, true2, X, y15(X, Y), y4(X, Y))
% 63.60/8.70 = { by axiom 14 (axiom_3_1) }
% 63.60/8.70 y4(X, Y)
% 63.60/8.70
% 63.60/8.70 Lemma 23: r2(y22(X, Y), y4(X, Y)) = true2.
% 63.60/8.70 Proof:
% 63.60/8.70 r2(y22(X, Y), y4(X, Y))
% 63.60/8.70 = { by lemma 20 R->L }
% 63.60/8.70 r2(y7(X, Y), y4(X, Y))
% 63.60/8.70 = { by axiom 15 (axiom_1a_2) }
% 63.60/8.70 true2
% 63.60/8.70
% 63.60/8.70 Goal 1 (twoplustwoeqfour): tuple(r1(X), r1(Y), r2(Z, W), r2(V, Z), r2(U, V), r2(X, U), r2(T, S), r2(Y, T), r3(S, S, W)) = tuple(true2, true2, true2, true2, true2, true2, true2, true2, true2).
% 63.60/8.70 The goal is true when:
% 63.60/8.70 X = y24
% 63.60/8.70 Y = y24
% 63.60/8.70 Z = y22(y21(y21(y24)), y21(y24))
% 63.60/8.70 W = y7(y21(y21(y24)), y21(y21(y24)))
% 63.60/8.70 V = y22(y21(y21(y24)), y24)
% 63.60/8.70 U = y21(y24)
% 63.60/8.70 T = y21(y24)
% 63.60/8.70 S = y21(y21(y24))
% 63.60/8.70
% 63.60/8.70 Proof:
% 63.60/8.70 tuple(r1(y24), r1(y24), r2(y22(y21(y21(y24)), y21(y24)), y7(y21(y21(y24)), y21(y21(y24)))), r2(y22(y21(y21(y24)), y24), y22(y21(y21(y24)), y21(y24))), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), r2(y21(y24), y21(y21(y24))), r2(y24, y21(y24)), r3(y21(y21(y24)), y21(y21(y24)), y7(y21(y21(y24)), y21(y21(y24)))))
% 63.60/8.70 = { by axiom 11 (axiom_1a_3) }
% 63.60/8.70 tuple(r1(y24), r1(y24), r2(y22(y21(y21(y24)), y21(y24)), y7(y21(y21(y24)), y21(y21(y24)))), r2(y22(y21(y21(y24)), y24), y22(y21(y21(y24)), y21(y24))), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), r2(y21(y24), y21(y21(y24))), r2(y24, y21(y24)), true2)
% 63.60/8.70 = { by lemma 20 }
% 63.60/8.70 tuple(r1(y24), r1(y24), r2(y22(y21(y21(y24)), y21(y24)), y22(y21(y21(y24)), y21(y21(y24)))), r2(y22(y21(y21(y24)), y24), y22(y21(y21(y24)), y21(y24))), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), r2(y21(y24), y21(y21(y24))), r2(y24, y21(y24)), true2)
% 63.60/8.70 = { by lemma 21 }
% 63.60/8.70 tuple(r1(y24), r1(y24), r2(y22(y21(y21(y24)), y21(y24)), y22(y21(y21(y24)), y21(y21(y24)))), r2(y22(y21(y21(y24)), y24), y22(y21(y21(y24)), y21(y24))), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), true2, r2(y24, y21(y24)), true2)
% 63.60/8.70 = { by lemma 22 }
% 63.60/8.70 tuple(r1(y24), r1(y24), r2(y22(y21(y21(y24)), y21(y24)), y4(y21(y21(y24)), y21(y24))), r2(y22(y21(y21(y24)), y24), y22(y21(y21(y24)), y21(y24))), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), true2, r2(y24, y21(y24)), true2)
% 63.60/8.70 = { by lemma 23 }
% 63.60/8.70 tuple(r1(y24), r1(y24), true2, r2(y22(y21(y21(y24)), y24), y22(y21(y21(y24)), y21(y24))), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), true2, r2(y24, y21(y24)), true2)
% 63.60/8.70 = { by lemma 21 }
% 63.60/8.70 tuple(r1(y24), r1(y24), true2, r2(y22(y21(y21(y24)), y24), y22(y21(y21(y24)), y21(y24))), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.70 = { by lemma 22 }
% 63.60/8.70 tuple(r1(y24), r1(y24), true2, r2(y22(y21(y21(y24)), y24), y4(y21(y21(y24)), y24)), r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.70 = { by lemma 23 }
% 63.60/8.70 tuple(r1(y24), r1(y24), true2, true2, r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.70 = { by lemma 19 }
% 63.60/8.70 tuple(r1(y24), true2, true2, true2, r2(y21(y24), y22(y21(y21(y24)), y24)), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.70 = { by axiom 4 (axiom_1_1) R->L }
% 63.60/8.70 tuple(r1(y24), true2, true2, true2, r2(y21(y24), y22(y21(y21(y24)), fresh10(true2, true2, y16(y21(y21(y24)))))), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.70 = { by axiom 2 (axiom_4a_1) R->L }
% 63.60/8.70 tuple(r1(y24), true2, true2, true2, r2(y21(y24), y22(y21(y21(y24)), fresh10(r1(y16(y21(y21(y24)))), true2, y16(y21(y21(y24)))))), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by axiom 8 (axiom_1_1) }
% 63.60/8.71 tuple(r1(y24), true2, true2, true2, r2(y21(y24), y22(y21(y21(y24)), y16(y21(y21(y24))))), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by axiom 18 (axiom_3_1) R->L }
% 63.60/8.71 tuple(r1(y24), true2, true2, true2, r2(y21(y24), fresh4(r3(y21(y21(y24)), y16(y21(y21(y24))), y21(y21(y24))), true2, y21(y21(y24)), y16(y21(y21(y24))), y21(y21(y24)))), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by axiom 1 (axiom_4a) R->L }
% 63.60/8.71 tuple(r1(y24), true2, true2, true2, r2(y21(y24), fresh4(r3(y21(y21(y24)), y16(y21(y21(y24))), y9(y21(y21(y24)))), true2, y21(y21(y24)), y16(y21(y21(y24))), y21(y21(y24)))), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by axiom 12 (axiom_4a_2) }
% 63.60/8.71 tuple(r1(y24), true2, true2, true2, r2(y21(y24), fresh4(true2, true2, y21(y21(y24)), y16(y21(y21(y24))), y21(y21(y24)))), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by axiom 14 (axiom_3_1) }
% 63.60/8.71 tuple(r1(y24), true2, true2, true2, r2(y21(y24), y21(y21(y24))), r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by lemma 21 }
% 63.60/8.71 tuple(r1(y24), true2, true2, true2, true2, r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by lemma 19 }
% 63.60/8.71 tuple(true2, true2, true2, true2, true2, r2(y24, y21(y24)), true2, true2, true2)
% 63.60/8.71 = { by lemma 21 }
% 63.60/8.71 tuple(true2, true2, true2, true2, true2, true2, true2, true2, true2)
% 63.60/8.71 % SZS output end Proof
% 63.60/8.71
% 63.60/8.71 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------