TSTP Solution File: NUM494+3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM494+3 : TPTP v8.1.2. Released v4.0.0.
% 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 : n006.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 11:56:37 EDT 2023
% Result : Theorem 266.66s 34.97s
% Output : Proof 266.66s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : NUM494+3 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Aug 25 15:19:07 EDT 2023
% 0.12/0.34 % CPUTime :
% 266.66/34.97 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 266.66/34.97
% 266.66/34.97 % SZS status Theorem
% 266.66/34.97
% 266.66/34.99 % SZS output start Proof
% 266.66/34.99 Take the following subset of the input axioms:
% 266.66/34.99 fof(mAddAsso, axiom, ![W0, W1, W2]: ((aNaturalNumber0(W0) & (aNaturalNumber0(W1) & aNaturalNumber0(W2))) => sdtpldt0(sdtpldt0(W0, W1), W2)=sdtpldt0(W0, sdtpldt0(W1, W2)))).
% 266.66/34.99 fof(mAddCanc, axiom, ![W0_2, W1_2, W2_2]: ((aNaturalNumber0(W0_2) & (aNaturalNumber0(W1_2) & aNaturalNumber0(W2_2))) => ((sdtpldt0(W0_2, W1_2)=sdtpldt0(W0_2, W2_2) | sdtpldt0(W1_2, W0_2)=sdtpldt0(W2_2, W0_2)) => W1_2=W2_2))).
% 266.66/34.99 fof(mAddComm, axiom, ![W0_2, W1_2]: ((aNaturalNumber0(W0_2) & aNaturalNumber0(W1_2)) => sdtpldt0(W0_2, W1_2)=sdtpldt0(W1_2, W0_2))).
% 266.66/34.99 fof(mSortsB, axiom, ![W0_2, W1_2]: ((aNaturalNumber0(W0_2) & aNaturalNumber0(W1_2)) => aNaturalNumber0(sdtpldt0(W0_2, W1_2)))).
% 266.66/34.99 fof(mSortsC, axiom, aNaturalNumber0(sz00)).
% 266.66/34.99 fof(m_AddZero, axiom, ![W0_2]: (aNaturalNumber0(W0_2) => (sdtpldt0(W0_2, sz00)=W0_2 & W0_2=sdtpldt0(sz00, W0_2)))).
% 266.66/34.99 fof(m__, conjecture, sdtpldt0(sdtpldt0(xr, xm), xp)!=sdtpldt0(sdtpldt0(xn, xm), xp) & (?[W0_2]: (aNaturalNumber0(W0_2) & sdtpldt0(sdtpldt0(sdtpldt0(xr, xm), xp), W0_2)=sdtpldt0(sdtpldt0(xn, xm), xp)) | sdtlseqdt0(sdtpldt0(sdtpldt0(xr, xm), xp), sdtpldt0(sdtpldt0(xn, xm), xp)))).
% 266.66/34.99 fof(m__1837, hypothesis, aNaturalNumber0(xn) & (aNaturalNumber0(xm) & aNaturalNumber0(xp))).
% 266.66/34.99 fof(m__1860, hypothesis, xp!=sz00 & (xp!=sz10 & (![W0_2]: ((aNaturalNumber0(W0_2) & (?[W1_2]: (aNaturalNumber0(W1_2) & xp=sdtasdt0(W0_2, W1_2)) | doDivides0(W0_2, xp))) => (W0_2=sz10 | W0_2=xp)) & (isPrime0(xp) & (?[W0_2]: (aNaturalNumber0(W0_2) & sdtasdt0(xn, xm)=sdtasdt0(xp, W0_2)) & doDivides0(xp, sdtasdt0(xn, xm))))))).
% 266.66/34.99 fof(m__1883, hypothesis, aNaturalNumber0(xr) & (sdtpldt0(xp, xr)=xn & xr=sdtmndt0(xn, xp))).
% 266.66/34.99
% 266.66/34.99 Now clausify the problem and encode Horn clauses using encoding 3 of
% 266.66/34.99 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 266.66/34.99 We repeatedly replace C & s=t => u=v by the two clauses:
% 266.66/34.99 fresh(y, y, x1...xn) = u
% 266.66/34.99 C => fresh(s, t, x1...xn) = v
% 266.66/34.99 where fresh is a fresh function symbol and x1..xn are the free
% 266.66/34.99 variables of u and v.
% 266.66/34.99 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 266.66/34.99 input problem has no model of domain size 1).
% 266.66/34.99
% 266.66/34.99 The encoding turns the above axioms into the following unit equations and goals:
% 266.66/34.99
% 266.66/34.99 Axiom 1 (m__1837_2): aNaturalNumber0(xp) = true2.
% 266.66/34.99 Axiom 2 (mSortsC): aNaturalNumber0(sz00) = true2.
% 266.66/34.99 Axiom 3 (m__1837): aNaturalNumber0(xn) = true2.
% 266.66/34.99 Axiom 4 (m__1837_1): aNaturalNumber0(xm) = true2.
% 266.66/35.00 Axiom 5 (m__1883_2): aNaturalNumber0(xr) = true2.
% 266.66/35.00 Axiom 6 (m__1883): sdtpldt0(xp, xr) = xn.
% 266.66/35.00 Axiom 7 (m_AddZero): fresh11(X, X, Y) = Y.
% 266.66/35.00 Axiom 8 (m__): fresh15(X, X) = sdtpldt0(sdtpldt0(xn, xm), xp).
% 266.66/35.00 Axiom 9 (m__): fresh14(X, X, Y) = sdtpldt0(sdtpldt0(xr, xm), xp).
% 266.66/35.00 Axiom 10 (mAddComm): fresh32(X, X, Y, Z) = sdtpldt0(Y, Z).
% 266.66/35.00 Axiom 11 (mAddComm): fresh31(X, X, Y, Z) = sdtpldt0(Z, Y).
% 266.66/35.00 Axiom 12 (mSortsB): fresh21(X, X, Y, Z) = aNaturalNumber0(sdtpldt0(Y, Z)).
% 266.66/35.00 Axiom 13 (mSortsB): fresh20(X, X, Y, Z) = true2.
% 266.66/35.00 Axiom 14 (m_AddZero): fresh11(aNaturalNumber0(X), true2, X) = sdtpldt0(X, sz00).
% 266.66/35.00 Axiom 15 (mAddCanc): fresh7(X, X, Y, Z) = Z.
% 266.66/35.00 Axiom 16 (mAddAsso): fresh112(X, X, Y, Z, W) = sdtpldt0(Y, sdtpldt0(Z, W)).
% 266.66/35.00 Axiom 17 (mAddCanc): fresh101(X, X, Y, Z, W) = Z.
% 266.66/35.00 Axiom 18 (mAddAsso): fresh33(X, X, Y, Z, W) = sdtpldt0(sdtpldt0(Y, Z), W).
% 266.66/35.00 Axiom 19 (mAddComm): fresh32(aNaturalNumber0(X), true2, Y, X) = fresh31(aNaturalNumber0(Y), true2, Y, X).
% 266.66/35.00 Axiom 20 (mSortsB): fresh21(aNaturalNumber0(X), true2, Y, X) = fresh20(aNaturalNumber0(Y), true2, Y, X).
% 266.66/35.00 Axiom 21 (mAddAsso): fresh111(X, X, Y, Z, W) = fresh112(aNaturalNumber0(Y), true2, Y, Z, W).
% 266.66/35.00 Axiom 22 (mAddCanc): fresh100(X, X, Y, Z, W) = fresh101(aNaturalNumber0(Y), true2, Y, Z, W).
% 266.66/35.00 Axiom 23 (mAddCanc): fresh99(X, X, Y, Z, W) = fresh100(aNaturalNumber0(Z), true2, Y, Z, W).
% 266.66/35.00 Axiom 24 (mAddAsso): fresh111(aNaturalNumber0(X), true2, Y, Z, X) = fresh33(aNaturalNumber0(Z), true2, Y, Z, X).
% 266.66/35.00 Axiom 25 (mAddCanc): fresh99(aNaturalNumber0(X), true2, Y, Z, X) = fresh7(sdtpldt0(Y, Z), sdtpldt0(Y, X), Z, X).
% 266.66/35.00 Axiom 26 (m__): fresh14(aNaturalNumber0(X), true2, X) = fresh15(sdtpldt0(sdtpldt0(sdtpldt0(xr, xm), xp), X), sdtpldt0(sdtpldt0(xn, xm), xp)).
% 266.66/35.00
% 266.66/35.00 Lemma 27: aNaturalNumber0(sdtpldt0(xn, xm)) = true2.
% 266.66/35.00 Proof:
% 266.66/35.00 aNaturalNumber0(sdtpldt0(xn, xm))
% 266.66/35.00 = { by axiom 12 (mSortsB) R->L }
% 266.66/35.00 fresh21(true2, true2, xn, xm)
% 266.66/35.00 = { by axiom 4 (m__1837_1) R->L }
% 266.66/35.00 fresh21(aNaturalNumber0(xm), true2, xn, xm)
% 266.66/35.00 = { by axiom 20 (mSortsB) }
% 266.66/35.00 fresh20(aNaturalNumber0(xn), true2, xn, xm)
% 266.66/35.00 = { by axiom 3 (m__1837) }
% 266.66/35.00 fresh20(true2, true2, xn, xm)
% 266.66/35.00 = { by axiom 13 (mSortsB) }
% 266.66/35.00 true2
% 266.66/35.00
% 266.66/35.00 Lemma 28: fresh14(X, X, Y) = sdtpldt0(xn, xm).
% 266.66/35.00 Proof:
% 266.66/35.00 fresh14(X, X, Y)
% 266.66/35.00 = { by axiom 9 (m__) }
% 266.66/35.00 sdtpldt0(sdtpldt0(xr, xm), xp)
% 266.66/35.00 = { by axiom 11 (mAddComm) R->L }
% 266.66/35.00 sdtpldt0(fresh31(true2, true2, xm, xr), xp)
% 266.66/35.01 = { by axiom 4 (m__1837_1) R->L }
% 266.66/35.01 sdtpldt0(fresh31(aNaturalNumber0(xm), true2, xm, xr), xp)
% 266.66/35.01 = { by axiom 19 (mAddComm) R->L }
% 266.66/35.01 sdtpldt0(fresh32(aNaturalNumber0(xr), true2, xm, xr), xp)
% 266.66/35.01 = { by axiom 5 (m__1883_2) }
% 266.66/35.01 sdtpldt0(fresh32(true2, true2, xm, xr), xp)
% 266.66/35.01 = { by axiom 10 (mAddComm) }
% 266.66/35.01 sdtpldt0(sdtpldt0(xm, xr), xp)
% 266.66/35.01 = { by axiom 18 (mAddAsso) R->L }
% 266.66/35.01 fresh33(true2, true2, xm, xr, xp)
% 266.66/35.01 = { by axiom 5 (m__1883_2) R->L }
% 266.66/35.01 fresh33(aNaturalNumber0(xr), true2, xm, xr, xp)
% 266.66/35.01 = { by axiom 24 (mAddAsso) R->L }
% 266.66/35.01 fresh111(aNaturalNumber0(xp), true2, xm, xr, xp)
% 266.66/35.01 = { by axiom 1 (m__1837_2) }
% 266.66/35.01 fresh111(true2, true2, xm, xr, xp)
% 266.66/35.01 = { by axiom 21 (mAddAsso) }
% 266.66/35.01 fresh112(aNaturalNumber0(xm), true2, xm, xr, xp)
% 266.66/35.01 = { by axiom 4 (m__1837_1) }
% 266.66/35.01 fresh112(true2, true2, xm, xr, xp)
% 266.66/35.01 = { by axiom 16 (mAddAsso) }
% 266.66/35.01 sdtpldt0(xm, sdtpldt0(xr, xp))
% 266.66/35.01 = { by axiom 11 (mAddComm) R->L }
% 266.66/35.01 sdtpldt0(xm, fresh31(true2, true2, xp, xr))
% 266.66/35.01 = { by axiom 1 (m__1837_2) R->L }
% 266.66/35.01 sdtpldt0(xm, fresh31(aNaturalNumber0(xp), true2, xp, xr))
% 266.66/35.01 = { by axiom 19 (mAddComm) R->L }
% 266.66/35.01 sdtpldt0(xm, fresh32(aNaturalNumber0(xr), true2, xp, xr))
% 266.66/35.01 = { by axiom 5 (m__1883_2) }
% 266.66/35.01 sdtpldt0(xm, fresh32(true2, true2, xp, xr))
% 266.66/35.01 = { by axiom 10 (mAddComm) }
% 266.66/35.01 sdtpldt0(xm, sdtpldt0(xp, xr))
% 266.66/35.01 = { by axiom 6 (m__1883) }
% 266.66/35.01 sdtpldt0(xm, xn)
% 266.66/35.01 = { by axiom 11 (mAddComm) R->L }
% 266.66/35.01 fresh31(true2, true2, xn, xm)
% 266.66/35.01 = { by axiom 3 (m__1837) R->L }
% 266.66/35.01 fresh31(aNaturalNumber0(xn), true2, xn, xm)
% 266.66/35.01 = { by axiom 19 (mAddComm) R->L }
% 266.66/35.01 fresh32(aNaturalNumber0(xm), true2, xn, xm)
% 266.66/35.01 = { by axiom 4 (m__1837_1) }
% 266.66/35.01 fresh32(true2, true2, xn, xm)
% 266.66/35.01 = { by axiom 10 (mAddComm) }
% 266.66/35.01 sdtpldt0(xn, xm)
% 266.66/35.01
% 266.66/35.01 Goal 1 (m__1860_4): xp = sz00.
% 266.66/35.01 Proof:
% 266.66/35.01 xp
% 266.66/35.01 = { by axiom 17 (mAddCanc) R->L }
% 266.66/35.01 fresh101(true2, true2, sdtpldt0(xn, xm), xp, sz00)
% 266.66/35.01 = { by lemma 27 R->L }
% 266.66/35.01 fresh101(aNaturalNumber0(sdtpldt0(xn, xm)), true2, sdtpldt0(xn, xm), xp, sz00)
% 266.66/35.01 = { by axiom 22 (mAddCanc) R->L }
% 266.66/35.01 fresh100(true2, true2, sdtpldt0(xn, xm), xp, sz00)
% 266.66/35.01 = { by axiom 1 (m__1837_2) R->L }
% 266.66/35.01 fresh100(aNaturalNumber0(xp), true2, sdtpldt0(xn, xm), xp, sz00)
% 266.66/35.01 = { by axiom 23 (mAddCanc) R->L }
% 266.66/35.01 fresh99(true2, true2, sdtpldt0(xn, xm), xp, sz00)
% 266.66/35.01 = { by axiom 2 (mSortsC) R->L }
% 266.66/35.01 fresh99(aNaturalNumber0(sz00), true2, sdtpldt0(xn, xm), xp, sz00)
% 266.66/35.01 = { by axiom 25 (mAddCanc) }
% 266.66/35.01 fresh7(sdtpldt0(sdtpldt0(xn, xm), xp), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by axiom 8 (m__) R->L }
% 266.66/35.01 fresh7(fresh15(fresh15(X, X), fresh15(X, X)), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by axiom 8 (m__) }
% 266.66/35.01 fresh7(fresh15(sdtpldt0(sdtpldt0(xn, xm), xp), fresh15(X, X)), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by lemma 28 R->L }
% 266.66/35.01 fresh7(fresh15(sdtpldt0(fresh14(Y, Y, Z), xp), fresh15(X, X)), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by axiom 8 (m__) }
% 266.66/35.01 fresh7(fresh15(sdtpldt0(fresh14(Y, Y, Z), xp), sdtpldt0(sdtpldt0(xn, xm), xp)), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by axiom 9 (m__) }
% 266.66/35.01 fresh7(fresh15(sdtpldt0(sdtpldt0(sdtpldt0(xr, xm), xp), xp), sdtpldt0(sdtpldt0(xn, xm), xp)), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by axiom 26 (m__) R->L }
% 266.66/35.01 fresh7(fresh14(aNaturalNumber0(xp), true2, xp), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by axiom 1 (m__1837_2) }
% 266.66/35.01 fresh7(fresh14(true2, true2, xp), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by lemma 28 }
% 266.66/35.01 fresh7(sdtpldt0(xn, xm), sdtpldt0(sdtpldt0(xn, xm), sz00), xp, sz00)
% 266.66/35.01 = { by axiom 14 (m_AddZero) R->L }
% 266.66/35.01 fresh7(sdtpldt0(xn, xm), fresh11(aNaturalNumber0(sdtpldt0(xn, xm)), true2, sdtpldt0(xn, xm)), xp, sz00)
% 266.66/35.01 = { by lemma 27 }
% 266.66/35.01 fresh7(sdtpldt0(xn, xm), fresh11(true2, true2, sdtpldt0(xn, xm)), xp, sz00)
% 266.66/35.01 = { by axiom 7 (m_AddZero) }
% 266.66/35.01 fresh7(sdtpldt0(xn, xm), sdtpldt0(xn, xm), xp, sz00)
% 266.66/35.01 = { by axiom 15 (mAddCanc) }
% 266.66/35.01 sz00
% 266.66/35.01 % SZS output end Proof
% 266.66/35.01
% 266.66/35.01 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------