TSTP Solution File: NUN067+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : NUN067+1 : 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 : n005.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 12:51:49 EDT 2023
% Result : Theorem 0.22s 0.49s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : NUN067+1 : TPTP v8.1.2. Released v7.3.0.
% 0.08/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n005.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Sun Aug 27 09:37:08 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.22/0.49 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.22/0.49
% 0.22/0.49 % SZS status Theorem
% 0.22/0.49
% 0.22/0.49 % SZS output start Proof
% 0.22/0.49 Take the following subset of the input axioms:
% 0.22/0.49 fof(axiom_1, axiom, ?[Y24]: ![X19]: ((id(X19, Y24) & r1(X19)) | (~r1(X19) & ~id(X19, Y24)))).
% 0.22/0.49 fof(axiom_2, axiom, ![X11]: ?[Y21]: ![X12]: ((id(X12, Y21) & r2(X11, X12)) | (~r2(X11, X12) & ~id(X12, Y21)))).
% 0.22/0.49 fof(axiom_6, axiom, ![X21, X22]: (~id(X21, X22) | id(X22, X21))).
% 0.22/0.49 fof(axiom_7a, axiom, ![X7, Y10]: (![Y20]: (~id(Y20, Y10) | ~r1(Y20)) | ~r2(X7, Y10))).
% 0.22/0.49 fof(axiom_8, axiom, ![X26, X27]: (~id(X26, X27) | ((~r1(X26) & ~r1(X27)) | (r1(X26) & r1(X27))))).
% 0.22/0.49 fof(nonzerosexistid, conjecture, ?[Y1]: ![Y2]: (~id(Y1, Y2) | ~r1(Y2))).
% 0.22/0.49
% 0.22/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.49 fresh(y, y, x1...xn) = u
% 0.22/0.49 C => fresh(s, t, x1...xn) = v
% 0.22/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.49 variables of u and v.
% 0.22/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.49 input problem has no model of domain size 1).
% 0.22/0.49
% 0.22/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.49
% 0.22/0.49 Axiom 1 (nonzerosexistid_1): r1(y2(X)) = true2.
% 0.22/0.49 Axiom 2 (nonzerosexistid): id(X, y2(X)) = true2.
% 0.22/0.49 Axiom 3 (axiom_1_1): fresh17(X, X, Y) = true2.
% 0.22/0.49 Axiom 4 (axiom_8_1): fresh3(X, X, Y) = true2.
% 0.22/0.49 Axiom 5 (axiom_1_1): fresh17(r1(X), true2, X) = id(X, y24).
% 0.22/0.49 Axiom 6 (axiom_2): fresh16(X, X, Y, Z) = true2.
% 0.22/0.49 Axiom 7 (axiom_6): fresh9(X, X, Y, Z) = true2.
% 0.22/0.49 Axiom 8 (axiom_8_1): fresh4(X, X, Y, Z) = r1(Y).
% 0.22/0.49 Axiom 9 (axiom_8_1): fresh4(r1(X), true2, Y, X) = fresh3(id(Y, X), true2, Y).
% 0.22/0.49 Axiom 10 (axiom_6): fresh9(id(X, Y), true2, X, Y) = id(Y, X).
% 0.22/0.49 Axiom 11 (axiom_2): fresh16(id(X, y21(Y)), true2, Y, X) = r2(Y, X).
% 0.22/0.49
% 0.22/0.49 Lemma 12: r1(X) = true2.
% 0.22/0.49 Proof:
% 0.22/0.49 r1(X)
% 0.22/0.49 = { by axiom 8 (axiom_8_1) R->L }
% 0.22/0.49 fresh4(true2, true2, X, y2(X))
% 0.22/0.49 = { by axiom 1 (nonzerosexistid_1) R->L }
% 0.22/0.49 fresh4(r1(y2(X)), true2, X, y2(X))
% 0.22/0.49 = { by axiom 9 (axiom_8_1) }
% 0.22/0.49 fresh3(id(X, y2(X)), true2, X)
% 0.22/0.50 = { by axiom 2 (nonzerosexistid) }
% 0.22/0.50 fresh3(true2, true2, X)
% 0.22/0.50 = { by axiom 4 (axiom_8_1) }
% 0.22/0.50 true2
% 0.22/0.50
% 0.22/0.50 Lemma 13: id(X, y24) = true2.
% 0.22/0.50 Proof:
% 0.22/0.50 id(X, y24)
% 0.22/0.50 = { by axiom 5 (axiom_1_1) R->L }
% 0.22/0.50 fresh17(r1(X), true2, X)
% 0.22/0.50 = { by lemma 12 }
% 0.22/0.50 fresh17(true2, true2, X)
% 0.22/0.50 = { by axiom 3 (axiom_1_1) }
% 0.22/0.50 true2
% 0.22/0.50
% 0.22/0.50 Goal 1 (axiom_7a): tuple(id(X, Y), r1(X), r2(Z, Y)) = tuple(true2, true2, true2).
% 0.22/0.50 The goal is true when:
% 0.22/0.50 X = X
% 0.22/0.50 Y = y24
% 0.22/0.50 Z = Y
% 0.22/0.50
% 0.22/0.50 Proof:
% 0.22/0.50 tuple(id(X, y24), r1(X), r2(Y, y24))
% 0.22/0.50 = { by axiom 11 (axiom_2) R->L }
% 0.22/0.50 tuple(id(X, y24), r1(X), fresh16(id(y24, y21(Y)), true2, Y, y24))
% 0.22/0.50 = { by axiom 10 (axiom_6) R->L }
% 0.22/0.50 tuple(id(X, y24), r1(X), fresh16(fresh9(id(y21(Y), y24), true2, y21(Y), y24), true2, Y, y24))
% 0.22/0.50 = { by lemma 13 }
% 0.22/0.50 tuple(id(X, y24), r1(X), fresh16(fresh9(true2, true2, y21(Y), y24), true2, Y, y24))
% 0.22/0.50 = { by axiom 7 (axiom_6) }
% 0.22/0.50 tuple(id(X, y24), r1(X), fresh16(true2, true2, Y, y24))
% 0.22/0.50 = { by axiom 6 (axiom_2) }
% 0.22/0.50 tuple(id(X, y24), r1(X), true2)
% 0.22/0.50 = { by lemma 13 }
% 0.22/0.50 tuple(true2, r1(X), true2)
% 0.22/0.50 = { by lemma 12 }
% 0.22/0.50 tuple(true2, true2, true2)
% 0.22/0.50 % SZS output end Proof
% 0.22/0.50
% 0.22/0.50 RESULT: Theorem (the conjecture is true).
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