TSTP Solution File: NUM450+6 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM450+6 : 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 : n026.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 11:56:21 EDT 2023
% Result : Theorem 4.27s 1.04s
% Output : Proof 4.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : NUM450+6 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n026.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 13:52:03 EDT 2023
% 0.13/0.34 % CPUTime :
% 4.27/1.04 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 4.27/1.04
% 4.27/1.04 % SZS status Theorem
% 4.27/1.04
% 4.27/1.04 % SZS output start Proof
% 4.27/1.04 Take the following subset of the input axioms:
% 4.73/1.07 fof(mComplement, definition, ![W0]: (aSubsetOf0(W0, cS1395) => ![W1]: (W1=stldt0(W0) <=> (aSet0(W1) & ![W2]: (aElementOf0(W2, W1) <=> (aInteger0(W2) & ~aElementOf0(W2, W0))))))).
% 4.73/1.07 fof(mDivisor, definition, ![W0_2]: (aInteger0(W0_2) => ![W1_2]: (aDivisorOf0(W1_2, W0_2) <=> (aInteger0(W1_2) & (W1_2!=sz00 & ?[W2_2]: (aInteger0(W2_2) & sdtasdt0(W1_2, W2_2)=W0_2)))))).
% 4.73/1.07 fof(mOpen, definition, ![W0_2]: (aSubsetOf0(W0_2, cS1395) => (isOpen0(W0_2) <=> ![W1_2]: (aElementOf0(W1_2, W0_2) => ?[W2_2]: (aInteger0(W2_2) & (W2_2!=sz00 & aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(W1_2, W2_2), W0_2))))))).
% 4.73/1.07 fof(mPrimeDivisor, axiom, ![W0_2]: (aInteger0(W0_2) => (?[W1_2]: (aDivisorOf0(W1_2, W0_2) & isPrime0(W1_2)) <=> (W0_2!=sz10 & W0_2!=smndt0(sz10))))).
% 4.73/1.07 fof(m__, conjecture, ?[W0_2]: (aInteger0(W0_2) & (W0_2!=sz00 & ((aSet0(szAzrzSzezqlpdtcmdtrp0(sz10, W0_2)) & ![W1_2]: ((aElementOf0(W1_2, szAzrzSzezqlpdtcmdtrp0(sz10, W0_2)) => (aInteger0(W1_2) & (?[W2_2]: (aInteger0(W2_2) & sdtasdt0(W0_2, W2_2)=sdtpldt0(W1_2, smndt0(sz10))) & (aDivisorOf0(W0_2, sdtpldt0(W1_2, smndt0(sz10))) & sdteqdtlpzmzozddtrp0(W1_2, sz10, W0_2))))) & ((aInteger0(W1_2) & (?[W2_2]: (aInteger0(W2_2) & sdtasdt0(W0_2, W2_2)=sdtpldt0(W1_2, smndt0(sz10))) | (aDivisorOf0(W0_2, sdtpldt0(W1_2, smndt0(sz10))) | sdteqdtlpzmzozddtrp0(W1_2, sz10, W0_2)))) => aElementOf0(W1_2, szAzrzSzezqlpdtcmdtrp0(sz10, W0_2))))) => ((aSet0(sbsmnsldt0(xS)) & ![W1_2]: (aElementOf0(W1_2, sbsmnsldt0(xS)) <=> (aInteger0(W1_2) & ?[W2_2]: (aElementOf0(W2_2, xS) & aElementOf0(W1_2, W2_2))))) => (![W1_2]: (aElementOf0(W1_2, stldt0(sbsmnsldt0(xS))) <=> (aInteger0(W1_2) & ~aElementOf0(W1_2, sbsmnsldt0(xS)))) => (![W1_2]: (aElementOf0(W1_2, szAzrzSzezqlpdtcmdtrp0(sz10, W0_2)) => aElementOf0(W1_2, stldt0(sbsmnsldt0(xS)))) | aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(sz10, W0_2), stldt0(sbsmnsldt0(xS)))))))))).
% 4.73/1.07 fof(m__2046, hypothesis, aSet0(xS) & (![W0_2]: ((aElementOf0(W0_2, xS) => ?[W1_2]: (aInteger0(W1_2) & (W1_2!=sz00 & (isPrime0(W1_2) & (aSet0(szAzrzSzezqlpdtcmdtrp0(sz00, W1_2)) & (![W2_2]: ((aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(sz00, W1_2)) => (aInteger0(W2_2) & (?[W3]: (aInteger0(W3) & sdtasdt0(W1_2, W3)=sdtpldt0(W2_2, smndt0(sz00))) & (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(sz00))) & sdteqdtlpzmzozddtrp0(W2_2, sz00, W1_2))))) & ((aInteger0(W2_2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2_2, smndt0(sz00))) | (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(sz00))) | sdteqdtlpzmzozddtrp0(W2_2, sz00, W1_2)))) => aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(sz00, W1_2)))) & szAzrzSzezqlpdtcmdtrp0(sz00, W1_2)=W0_2)))))) & (?[W1_2]: (aInteger0(W1_2) & (W1_2!=sz00 & (isPrime0(W1_2) & ((aSet0(szAzrzSzezqlpdtcmdtrp0(sz00, W1_2)) & ![W2_2]: ((aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(sz00, W1_2)) => (aInteger0(W2_2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2_2, smndt0(sz00))) & (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(sz00))) & sdteqdtlpzmzozddtrp0(W2_2, sz00, W1_2))))) & ((aInteger0(W2_2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2_2, smndt0(sz00))) | (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(sz00))) | sdteqdtlpzmzozddtrp0(W2_2, sz00, W1_2)))) => aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(sz00, W1_2))))) => szAzrzSzezqlpdtcmdtrp0(sz00, W1_2)=W0_2)))) => aElementOf0(W0_2, xS))) & xS=cS2043)).
% 4.73/1.07 fof(m__2079, hypothesis, aSet0(sbsmnsldt0(xS)) & (![W0_2]: (aElementOf0(W0_2, sbsmnsldt0(xS)) <=> (aInteger0(W0_2) & ?[W1_2]: (aElementOf0(W1_2, xS) & aElementOf0(W0_2, W1_2)))) & (aSet0(stldt0(sbsmnsldt0(xS))) & (![W0_2]: (aElementOf0(W0_2, stldt0(sbsmnsldt0(xS))) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, sbsmnsldt0(xS)))) & (![W0_2]: (aElementOf0(W0_2, stldt0(sbsmnsldt0(xS))) <=> (W0_2=sz10 | W0_2=smndt0(sz10))) & stldt0(sbsmnsldt0(xS))=cS2076))))).
% 4.73/1.07 fof(m__2144, hypothesis, aSet0(sbsmnsldt0(xS)) & (![W0_2]: (aElementOf0(W0_2, sbsmnsldt0(xS)) <=> (aInteger0(W0_2) & ?[W1_2]: (aElementOf0(W1_2, xS) & aElementOf0(W0_2, W1_2)))) & (![W0_2]: (aElementOf0(W0_2, stldt0(sbsmnsldt0(xS))) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, sbsmnsldt0(xS)))) & (![W0_2]: (aElementOf0(W0_2, stldt0(sbsmnsldt0(xS))) => ?[W1_2]: (aInteger0(W1_2) & (W1_2!=sz00 & (aSet0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) & (![W2_2]: ((aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) => (aInteger0(W2_2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2_2, smndt0(W0_2))) & (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(W0_2))) & sdteqdtlpzmzozddtrp0(W2_2, W0_2, W1_2))))) & ((aInteger0(W2_2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2_2, smndt0(W0_2))) | (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(W0_2))) | sdteqdtlpzmzozddtrp0(W2_2, W0_2, W1_2)))) => aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)))) & (![W2_2]: (aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) => aElementOf0(W2_2, stldt0(sbsmnsldt0(xS)))) & aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2), stldt0(sbsmnsldt0(xS))))))))) & (isOpen0(stldt0(sbsmnsldt0(xS))) & (isClosed0(sbsmnsldt0(xS)) & (aSet0(sbsmnsldt0(xS)) & (![W0_2]: (aElementOf0(W0_2, sbsmnsldt0(xS)) <=> (aInteger0(W0_2) & ?[W1_2]: (aElementOf0(W1_2, xS) & aElementOf0(W0_2, W1_2)))) & (![W0_2]: (aElementOf0(W0_2, stldt0(sbsmnsldt0(xS))) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, sbsmnsldt0(xS)))) & ![W0_2]: (aElementOf0(W0_2, stldt0(sbsmnsldt0(xS))) => ?[W1_2]: (aInteger0(W1_2) & (W1_2!=sz00 & (aSet0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) & (![W2_2]: ((aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) => (aInteger0(W2_2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2_2, smndt0(W0_2))) & (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(W0_2))) & sdteqdtlpzmzozddtrp0(W2_2, W0_2, W1_2))))) & ((aInteger0(W2_2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2_2, smndt0(W0_2))) | (aDivisorOf0(W1_2, sdtpldt0(W2_2, smndt0(W0_2))) | sdteqdtlpzmzozddtrp0(W2_2, W0_2, W1_2)))) => aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)))) & (![W2_2]: (aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) => aElementOf0(W2_2, stldt0(sbsmnsldt0(xS)))) & aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2), stldt0(sbsmnsldt0(xS)))))))))))))))))).
% 4.73/1.07
% 4.73/1.07 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.73/1.07 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.73/1.07 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.73/1.07 fresh(y, y, x1...xn) = u
% 4.73/1.07 C => fresh(s, t, x1...xn) = v
% 4.73/1.07 where fresh is a fresh function symbol and x1..xn are the free
% 4.73/1.07 variables of u and v.
% 4.73/1.07 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.73/1.07 input problem has no model of domain size 1).
% 4.73/1.07
% 4.73/1.07 The encoding turns the above axioms into the following unit equations and goals:
% 4.73/1.07
% 4.73/1.07 Axiom 1 (m__2079): stldt0(sbsmnsldt0(xS)) = cS2076.
% 4.73/1.07 Axiom 2 (m___19): fresh(X, X, Y) = Y.
% 4.73/1.07 Axiom 3 (m__2079_3): fresh63(X, X, Y) = true2.
% 4.73/1.07 Axiom 4 (m__2144_20): fresh49(X, X, Y) = true2.
% 4.73/1.07 Axiom 5 (m__2144_22): fresh47(X, X, Y) = true2.
% 4.73/1.07 Axiom 6 (m___19): fresh10(X, X, Y) = sz00.
% 4.73/1.07 Axiom 7 (m__2079_3): fresh63(X, sz10, X) = aElementOf0(X, stldt0(sbsmnsldt0(xS))).
% 4.73/1.07 Axiom 8 (m__2144_20): fresh49(aElementOf0(X, stldt0(sbsmnsldt0(xS))), true2, X) = aInteger0(w1_3(X)).
% 4.73/1.07 Axiom 9 (m__2144_22): fresh47(aElementOf0(X, stldt0(sbsmnsldt0(xS))), true2, X) = aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(X, w1_3(X)), stldt0(sbsmnsldt0(xS))).
% 4.73/1.07 Axiom 10 (m___19): fresh(aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(sz10, X), stldt0(sbsmnsldt0(xS))), true2, X) = fresh10(aInteger0(X), true2, X).
% 4.73/1.07
% 4.73/1.07 Lemma 11: aElementOf0(sz10, cS2076) = true2.
% 4.73/1.07 Proof:
% 4.73/1.07 aElementOf0(sz10, cS2076)
% 4.73/1.07 = { by axiom 1 (m__2079) R->L }
% 4.73/1.07 aElementOf0(sz10, stldt0(sbsmnsldt0(xS)))
% 4.73/1.07 = { by axiom 7 (m__2079_3) R->L }
% 4.73/1.07 fresh63(sz10, sz10, sz10)
% 4.73/1.07 = { by axiom 3 (m__2079_3) }
% 4.73/1.07 true2
% 4.73/1.07
% 4.73/1.07 Goal 1 (m__2144_3): tuple4(w1_3(X), aElementOf0(X, stldt0(sbsmnsldt0(xS)))) = tuple4(sz00, true2).
% 4.73/1.07 The goal is true when:
% 4.73/1.07 X = sz10
% 4.73/1.07
% 4.73/1.07 Proof:
% 4.73/1.07 tuple4(w1_3(sz10), aElementOf0(sz10, stldt0(sbsmnsldt0(xS))))
% 4.73/1.07 = { by axiom 1 (m__2079) }
% 4.73/1.07 tuple4(w1_3(sz10), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 2 (m___19) R->L }
% 4.73/1.07 tuple4(fresh(true2, true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 5 (m__2144_22) R->L }
% 4.73/1.07 tuple4(fresh(fresh47(true2, true2, sz10), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by lemma 11 R->L }
% 4.73/1.07 tuple4(fresh(fresh47(aElementOf0(sz10, cS2076), true2, sz10), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 1 (m__2079) R->L }
% 4.73/1.07 tuple4(fresh(fresh47(aElementOf0(sz10, stldt0(sbsmnsldt0(xS))), true2, sz10), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 9 (m__2144_22) }
% 4.73/1.07 tuple4(fresh(aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(sz10, w1_3(sz10)), stldt0(sbsmnsldt0(xS))), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 10 (m___19) }
% 4.73/1.07 tuple4(fresh10(aInteger0(w1_3(sz10)), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 8 (m__2144_20) R->L }
% 4.73/1.07 tuple4(fresh10(fresh49(aElementOf0(sz10, stldt0(sbsmnsldt0(xS))), true2, sz10), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 1 (m__2079) }
% 4.73/1.07 tuple4(fresh10(fresh49(aElementOf0(sz10, cS2076), true2, sz10), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by lemma 11 }
% 4.73/1.07 tuple4(fresh10(fresh49(true2, true2, sz10), true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 4 (m__2144_20) }
% 4.73/1.07 tuple4(fresh10(true2, true2, w1_3(sz10)), aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by axiom 6 (m___19) }
% 4.73/1.07 tuple4(sz00, aElementOf0(sz10, cS2076))
% 4.73/1.07 = { by lemma 11 }
% 4.73/1.07 tuple4(sz00, true2)
% 4.73/1.07 % SZS output end Proof
% 4.73/1.07
% 4.73/1.07 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------