TSTP Solution File: NUN073+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : NUN073+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 : 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 12:51:51 EDT 2023
% Result : Theorem 0.13s 0.43s
% 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 : NUN073+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 : n016.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 09:59:28 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.13/0.43 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.13/0.43
% 0.13/0.43 % SZS status Theorem
% 0.13/0.43
% 0.13/0.43 % SZS output start Proof
% 0.13/0.44 Take the following subset of the input axioms:
% 0.13/0.44 fof(axiom_1, axiom, ?[Y24]: ![X19]: ((~r1(X19) & X19!=Y24) | (r1(X19) & X19=Y24))).
% 0.13/0.44 fof(axiom_2, axiom, ![X11]: ?[Y21]: ![X12]: ((~r2(X11, X12) & X12!=Y21) | (r2(X11, X12) & X12=Y21))).
% 0.13/0.44 fof(axiom_3a, axiom, ![X3, X10]: (![Y12]: (![Y13]: (~r2(X3, Y13) | Y13!=Y12) | ~r2(X10, Y12)) | X3=X10)).
% 0.13/0.44 fof(axiom_7a, axiom, ![X7, Y10]: (![Y20]: (~r1(Y20) | Y20!=Y10) | ~r2(X7, Y10))).
% 0.13/0.44 fof(oneuneqtwo, conjecture, ![Y1]: (![Y2]: (![Y4]: (~r1(Y4) | ~r2(Y4, Y2)) | Y2!=Y1) | ![Y3]: (![Y5]: (~r1(Y5) | ~r2(Y5, Y3)) | ~r2(Y3, Y1)))).
% 0.13/0.44
% 0.13/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.44 fresh(y, y, x1...xn) = u
% 0.13/0.44 C => fresh(s, t, x1...xn) = v
% 0.13/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.44 variables of u and v.
% 0.13/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.44 input problem has no model of domain size 1).
% 0.13/0.44
% 0.13/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.44
% 0.13/0.44 Axiom 1 (oneuneqtwo): y2 = y1.
% 0.13/0.44 Axiom 2 (oneuneqtwo_1): r1(y4) = true2.
% 0.13/0.44 Axiom 3 (oneuneqtwo_2): r1(y5) = true2.
% 0.13/0.44 Axiom 4 (oneuneqtwo_3): r2(y4, y2) = true2.
% 0.13/0.44 Axiom 5 (oneuneqtwo_4): r2(y3, y1) = true2.
% 0.13/0.44 Axiom 6 (oneuneqtwo_5): r2(y5, y3) = true2.
% 0.13/0.44 Axiom 7 (axiom_1_1): fresh10(X, X, Y) = y24.
% 0.13/0.44 Axiom 8 (axiom_1): fresh9(X, X, Y) = true2.
% 0.13/0.44 Axiom 9 (axiom_1): fresh9(X, y24, X) = r1(X).
% 0.20/0.44 Axiom 10 (axiom_3a): fresh(X, X, Y, Z) = Z.
% 0.20/0.44 Axiom 11 (axiom_1_1): fresh10(r1(X), true2, X) = X.
% 0.20/0.44 Axiom 12 (axiom_2_1): fresh5(X, X, Y, Z) = Z.
% 0.20/0.44 Axiom 13 (axiom_3a): fresh2(X, X, Y, Z, W) = Y.
% 0.20/0.44 Axiom 14 (axiom_2_1): fresh5(r2(X, Y), true2, X, Y) = y21(X).
% 0.20/0.44 Axiom 15 (axiom_3a): fresh2(r2(X, Y), true2, Z, X, Y) = fresh(r2(Z, Y), true2, Z, X).
% 0.20/0.44
% 0.20/0.44 Lemma 16: y4 = y24.
% 0.20/0.44 Proof:
% 0.20/0.44 y4
% 0.20/0.44 = { by axiom 11 (axiom_1_1) R->L }
% 0.20/0.44 fresh10(r1(y4), true2, y4)
% 0.20/0.44 = { by axiom 2 (oneuneqtwo_1) }
% 0.20/0.44 fresh10(true2, true2, y4)
% 0.20/0.44 = { by axiom 7 (axiom_1_1) }
% 0.20/0.44 y24
% 0.20/0.44
% 0.20/0.44 Lemma 17: r2(y4, y1) = true2.
% 0.20/0.44 Proof:
% 0.20/0.44 r2(y4, y1)
% 0.20/0.44 = { by axiom 1 (oneuneqtwo) R->L }
% 0.20/0.44 r2(y4, y2)
% 0.20/0.44 = { by axiom 4 (oneuneqtwo_3) }
% 0.20/0.44 true2
% 0.20/0.44
% 0.20/0.44 Goal 1 (axiom_7a): tuple(r1(X), r2(Y, X)) = tuple(true2, true2).
% 0.20/0.44 The goal is true when:
% 0.20/0.44 X = y24
% 0.20/0.44 Y = y24
% 0.20/0.44
% 0.20/0.44 Proof:
% 0.20/0.44 tuple(r1(y24), r2(y24, y24))
% 0.20/0.44 = { by lemma 16 R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, y24))
% 0.20/0.44 = { by axiom 10 (axiom_3a) R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh(true2, true2, y3, y24)))
% 0.20/0.44 = { by axiom 5 (oneuneqtwo_4) R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh(r2(y3, y1), true2, y3, y24)))
% 0.20/0.44 = { by lemma 16 R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh(r2(y3, y1), true2, y3, y4)))
% 0.20/0.44 = { by axiom 15 (axiom_3a) R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh2(r2(y4, y1), true2, y3, y4, y1)))
% 0.20/0.44 = { by lemma 17 }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh2(true2, true2, y3, y4, y1)))
% 0.20/0.44 = { by axiom 13 (axiom_3a) }
% 0.20/0.44 tuple(r1(y24), r2(y4, y3))
% 0.20/0.44 = { by axiom 12 (axiom_2_1) R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh5(true2, true2, y5, y3)))
% 0.20/0.44 = { by axiom 6 (oneuneqtwo_5) R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh5(r2(y5, y3), true2, y5, y3)))
% 0.20/0.44 = { by axiom 14 (axiom_2_1) }
% 0.20/0.44 tuple(r1(y24), r2(y4, y21(y5)))
% 0.20/0.44 = { by axiom 11 (axiom_1_1) R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, y21(fresh10(r1(y5), true2, y5))))
% 0.20/0.44 = { by axiom 3 (oneuneqtwo_2) }
% 0.20/0.44 tuple(r1(y24), r2(y4, y21(fresh10(true2, true2, y5))))
% 0.20/0.44 = { by axiom 7 (axiom_1_1) }
% 0.20/0.44 tuple(r1(y24), r2(y4, y21(y24)))
% 0.20/0.44 = { by lemma 16 R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, y21(y4)))
% 0.20/0.44 = { by axiom 14 (axiom_2_1) R->L }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh5(r2(y4, y1), true2, y4, y1)))
% 0.20/0.44 = { by lemma 17 }
% 0.20/0.44 tuple(r1(y24), r2(y4, fresh5(true2, true2, y4, y1)))
% 0.20/0.44 = { by axiom 12 (axiom_2_1) }
% 0.20/0.44 tuple(r1(y24), r2(y4, y1))
% 0.20/0.44 = { by lemma 17 }
% 0.20/0.44 tuple(r1(y24), true2)
% 0.20/0.44 = { by axiom 9 (axiom_1) R->L }
% 0.20/0.44 tuple(fresh9(y24, y24, y24), true2)
% 0.20/0.44 = { by axiom 8 (axiom_1) }
% 0.20/0.44 tuple(true2, true2)
% 0.20/0.44 % SZS output end Proof
% 0.20/0.44
% 0.20/0.44 RESULT: Theorem (the conjecture is true).
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