TSTP Solution File: NUM441+6 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM441+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 : n011.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 11:56:18 EDT 2023
% Result : Theorem 3.89s 0.88s
% Output : Proof 3.89s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : NUM441+6 : TPTP v8.1.2. Released v4.0.0.
% 0.14/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.36 % Computer : n011.cluster.edu
% 0.16/0.36 % Model : x86_64 x86_64
% 0.16/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36 % Memory : 8042.1875MB
% 0.16/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36 % CPULimit : 300
% 0.16/0.36 % WCLimit : 300
% 0.16/0.36 % DateTime : Fri Aug 25 15:20:23 EDT 2023
% 0.16/0.36 % CPUTime :
% 3.89/0.88 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 3.89/0.88
% 3.89/0.88 % SZS status Theorem
% 3.89/0.88
% 3.89/0.88 % SZS output start Proof
% 3.89/0.88 Take the following subset of the input axioms:
% 3.89/0.90 fof(mInterOpen, axiom, ![W0, W1]: ((aSubsetOf0(W0, cS1395) & (aSubsetOf0(W1, cS1395) & (isOpen0(W0) & isOpen0(W1)))) => isOpen0(sdtslmnbsdt0(W0, W1)))).
% 3.89/0.90 fof(m__, conjecture, (aSet0(sdtbsmnsldt0(xA, xB)) & ![W0_2]: (aElementOf0(W0_2, sdtbsmnsldt0(xA, xB)) <=> (aInteger0(W0_2) & (aElementOf0(W0_2, xA) | aElementOf0(W0_2, xB))))) => (((aSet0(stldt0(sdtbsmnsldt0(xA, xB))) & ![W0_2]: (aElementOf0(W0_2, stldt0(sdtbsmnsldt0(xA, xB))) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, sdtbsmnsldt0(xA, xB))))) => (![W0_2]: (aElementOf0(W0_2, stldt0(sdtbsmnsldt0(xA, xB))) => ?[W1_2]: (aInteger0(W1_2) & (W1_2!=sz00 & ((aSet0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) & ![W2]: ((aElementOf0(W2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) => (aInteger0(W2) & (?[W3]: (aInteger0(W3) & sdtasdt0(W1_2, W3)=sdtpldt0(W2, smndt0(W0_2))) & (aDivisorOf0(W1_2, sdtpldt0(W2, smndt0(W0_2))) & sdteqdtlpzmzozddtrp0(W2, W0_2, W1_2))))) & ((aInteger0(W2) & (?[W3_2]: (aInteger0(W3_2) & sdtasdt0(W1_2, W3_2)=sdtpldt0(W2, smndt0(W0_2))) | (aDivisorOf0(W1_2, sdtpldt0(W2, smndt0(W0_2))) | sdteqdtlpzmzozddtrp0(W2, W0_2, W1_2)))) => aElementOf0(W2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2))))) => (![W2_2]: (aElementOf0(W2_2, szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2)) => aElementOf0(W2_2, stldt0(sdtbsmnsldt0(xA, xB)))) | aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2), stldt0(sdtbsmnsldt0(xA, xB)))))))) | isOpen0(stldt0(sdtbsmnsldt0(xA, xB))))) | isClosed0(sdtbsmnsldt0(xA, xB)))).
% 3.89/0.90 fof(m__1826, hypothesis, aSet0(cS1395) & (![W0_2]: (aElementOf0(W0_2, cS1395) <=> aInteger0(W0_2)) & (aSet0(xA) & (![W0_2]: (aElementOf0(W0_2, xA) => aElementOf0(W0_2, cS1395)) & (aSubsetOf0(xA, cS1395) & (aSet0(cS1395) & (![W0_2]: (aElementOf0(W0_2, cS1395) <=> aInteger0(W0_2)) & (aSet0(xB) & (![W0_2]: (aElementOf0(W0_2, xB) => aElementOf0(W0_2, cS1395)) & (aSubsetOf0(xB, cS1395) & (aSet0(stldt0(xA)) & (![W0_2]: (aElementOf0(W0_2, stldt0(xA)) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, xA))) & (![W0_2]: (aElementOf0(W0_2, stldt0(xA)) => ?[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(xA))) & aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2), stldt0(xA)))))))) & (isOpen0(stldt0(xA)) & (isClosed0(xA) & (aSet0(stldt0(xB)) & (![W0_2]: (aElementOf0(W0_2, stldt0(xB)) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, xB))) & (![W0_2]: (aElementOf0(W0_2, stldt0(xB)) => ?[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(xB))) & aSubsetOf0(szAzrzSzezqlpdtcmdtrp0(W0_2, W1_2), stldt0(xB)))))))) & (isOpen0(stldt0(xB)) & isClosed0(xB)))))))))))))))))))).
% 3.89/0.90 fof(m__1883, hypothesis, aSet0(stldt0(xA)) & (![W0_2]: (aElementOf0(W0_2, stldt0(xA)) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, xA))) & (aSet0(cS1395) & (![W0_2]: (aElementOf0(W0_2, cS1395) <=> aInteger0(W0_2)) & (![W0_2]: (aElementOf0(W0_2, stldt0(xA)) => aElementOf0(W0_2, cS1395)) & (aSubsetOf0(stldt0(xA), cS1395) & (aSet0(stldt0(xB)) & (![W0_2]: (aElementOf0(W0_2, stldt0(xB)) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, xB))) & (aSet0(cS1395) & (![W0_2]: (aElementOf0(W0_2, cS1395) <=> aInteger0(W0_2)) & (![W0_2]: (aElementOf0(W0_2, stldt0(xB)) => aElementOf0(W0_2, cS1395)) & (aSubsetOf0(stldt0(xB), cS1395) & (aSet0(sdtbsmnsldt0(xA, xB)) & (![W0_2]: (aElementOf0(W0_2, sdtbsmnsldt0(xA, xB)) <=> (aInteger0(W0_2) & (aElementOf0(W0_2, xA) | aElementOf0(W0_2, xB)))) & (aSet0(stldt0(sdtbsmnsldt0(xA, xB))) & (![W0_2]: (aElementOf0(W0_2, stldt0(sdtbsmnsldt0(xA, xB))) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, sdtbsmnsldt0(xA, xB)))) & (![W0_2]: (aElementOf0(W0_2, stldt0(xA)) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, xA))) & (![W0_2]: (aElementOf0(W0_2, stldt0(xB)) <=> (aInteger0(W0_2) & ~aElementOf0(W0_2, xB))) & (![W0_2]: (aElementOf0(W0_2, stldt0(sdtbsmnsldt0(xA, xB))) <=> (aInteger0(W0_2) & (aElementOf0(W0_2, stldt0(xA)) & aElementOf0(W0_2, stldt0(xB))))) & stldt0(sdtbsmnsldt0(xA, xB))=sdtslmnbsdt0(stldt0(xA), stldt0(xB))))))))))))))))))))).
% 3.89/0.90
% 3.89/0.90 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.89/0.90 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.89/0.90 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.89/0.90 fresh(y, y, x1...xn) = u
% 3.89/0.90 C => fresh(s, t, x1...xn) = v
% 3.89/0.90 where fresh is a fresh function symbol and x1..xn are the free
% 3.89/0.90 variables of u and v.
% 3.89/0.90 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.89/0.90 input problem has no model of domain size 1).
% 3.89/0.90
% 3.89/0.90 The encoding turns the above axioms into the following unit equations and goals:
% 3.89/0.90
% 3.89/0.90 Axiom 1 (m__1826_7): isOpen0(stldt0(xA)) = true2.
% 3.89/0.90 Axiom 2 (m__1826_8): isOpen0(stldt0(xB)) = true2.
% 3.89/0.90 Axiom 3 (m__1883_6): aSubsetOf0(stldt0(xA), cS1395) = true2.
% 3.89/0.90 Axiom 4 (m__1883_7): aSubsetOf0(stldt0(xB), cS1395) = true2.
% 3.89/0.90 Axiom 5 (m__1883): stldt0(sdtbsmnsldt0(xA, xB)) = sdtslmnbsdt0(stldt0(xA), stldt0(xB)).
% 3.89/0.90 Axiom 6 (mInterOpen): fresh162(X, X, Y, Z) = true2.
% 3.89/0.90 Axiom 7 (mInterOpen): fresh112(X, X, Y, Z) = isOpen0(sdtslmnbsdt0(Y, Z)).
% 3.89/0.90 Axiom 8 (mInterOpen): fresh160(X, X, Y, Z) = fresh161(isOpen0(Y), true2, Y, Z).
% 3.89/0.90 Axiom 9 (mInterOpen): fresh161(X, X, Y, Z) = fresh162(aSubsetOf0(Y, cS1395), true2, Y, Z).
% 3.89/0.90 Axiom 10 (mInterOpen): fresh160(isOpen0(X), true2, Y, X) = fresh112(aSubsetOf0(X, cS1395), true2, Y, X).
% 3.89/0.90
% 3.89/0.90 Goal 1 (m___22): isOpen0(stldt0(sdtbsmnsldt0(xA, xB))) = true2.
% 3.89/0.90 Proof:
% 3.89/0.90 isOpen0(stldt0(sdtbsmnsldt0(xA, xB)))
% 3.89/0.90 = { by axiom 5 (m__1883) }
% 3.89/0.90 isOpen0(sdtslmnbsdt0(stldt0(xA), stldt0(xB)))
% 3.89/0.90 = { by axiom 7 (mInterOpen) R->L }
% 3.89/0.90 fresh112(true2, true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 4 (m__1883_7) R->L }
% 3.89/0.90 fresh112(aSubsetOf0(stldt0(xB), cS1395), true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 10 (mInterOpen) R->L }
% 3.89/0.90 fresh160(isOpen0(stldt0(xB)), true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 2 (m__1826_8) }
% 3.89/0.90 fresh160(true2, true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 8 (mInterOpen) }
% 3.89/0.90 fresh161(isOpen0(stldt0(xA)), true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 1 (m__1826_7) }
% 3.89/0.90 fresh161(true2, true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 9 (mInterOpen) }
% 3.89/0.90 fresh162(aSubsetOf0(stldt0(xA), cS1395), true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 3 (m__1883_6) }
% 3.89/0.90 fresh162(true2, true2, stldt0(xA), stldt0(xB))
% 3.89/0.90 = { by axiom 6 (mInterOpen) }
% 3.89/0.90 true2
% 3.89/0.90 % SZS output end Proof
% 3.89/0.90
% 3.89/0.90 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------