TSTP Solution File: RNG086+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG086+1 : 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 : n004.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 13:59:12 EDT 2023
% Result : Theorem 211.88s 27.66s
% Output : Proof 212.70s
% 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 : RNG086+1 : 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
% 0.14/0.34 % Computer : n004.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sun Aug 27 01:56:52 EDT 2023
% 0.14/0.35 % CPUTime :
% 211.88/27.66 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 211.88/27.66
% 211.88/27.66 % SZS status Theorem
% 211.88/27.66
% 211.88/27.66 % SZS output start Proof
% 211.88/27.66 Take the following subset of the input axioms:
% 211.88/27.66 fof(mDefIdeal, definition, ![W0]: (aIdeal0(W0) <=> (aSet0(W0) & ![W1]: (aElementOf0(W1, W0) => (![W2]: (aElementOf0(W2, W0) => aElementOf0(sdtpldt0(W1, W2), W0)) & ![W2_2]: (aElement0(W2_2) => aElementOf0(sdtasdt0(W2_2, W1), W0))))))).
% 211.88/27.66 fof(mDefSSum, definition, ![W0_2, W1_2]: ((aSet0(W0_2) & aSet0(W1_2)) => ![W2_2]: (W2_2=sdtpldt1(W0_2, W1_2) <=> (aSet0(W2_2) & ![W3]: (aElementOf0(W3, W2_2) <=> ?[W4, W5]: (aElementOf0(W4, W0_2) & (aElementOf0(W5, W1_2) & sdtpldt0(W4, W5)=W3))))))).
% 211.88/27.66 fof(m__, conjecture, ?[W0_2, W1_2]: (aElementOf0(W0_2, xI) & (aElementOf0(W1_2, xJ) & xx=sdtpldt0(W0_2, W1_2)))).
% 211.88/27.66 fof(m__870, hypothesis, aIdeal0(xI) & aIdeal0(xJ)).
% 211.88/27.66 fof(m__901, hypothesis, aElementOf0(xx, sdtpldt1(xI, xJ)) & (aElementOf0(xy, sdtpldt1(xI, xJ)) & aElement0(xz))).
% 211.88/27.66
% 211.88/27.66 Now clausify the problem and encode Horn clauses using encoding 3 of
% 211.88/27.66 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 211.88/27.66 We repeatedly replace C & s=t => u=v by the two clauses:
% 211.88/27.66 fresh(y, y, x1...xn) = u
% 211.88/27.66 C => fresh(s, t, x1...xn) = v
% 211.88/27.66 where fresh is a fresh function symbol and x1..xn are the free
% 211.88/27.66 variables of u and v.
% 211.88/27.66 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 211.88/27.66 input problem has no model of domain size 1).
% 211.88/27.66
% 211.88/27.66 The encoding turns the above axioms into the following unit equations and goals:
% 211.88/27.66
% 211.88/27.66 Axiom 1 (m__870): aIdeal0(xI) = true2.
% 211.88/27.66 Axiom 2 (m__870_1): aIdeal0(xJ) = true2.
% 211.88/27.66 Axiom 3 (mDefIdeal_7): fresh29(X, X, Y) = true2.
% 211.88/27.66 Axiom 4 (m__901_1): aElementOf0(xx, sdtpldt1(xI, xJ)) = true2.
% 211.88/27.66 Axiom 5 (mDefIdeal_7): fresh29(aIdeal0(X), true2, X) = aSet0(X).
% 211.88/27.66 Axiom 6 (mDefSSum_6): fresh(X, X, Y, Z, W) = W.
% 211.88/27.66 Axiom 7 (mDefSSum_1): fresh66(X, X, Y, Z, W) = true2.
% 211.88/27.66 Axiom 8 (mDefSSum_7): fresh21(X, X, Y, Z, W) = true2.
% 211.88/27.66 Axiom 9 (mDefSSum_8): fresh20(X, X, Y, Z, W) = true2.
% 211.88/27.67 Axiom 10 (mDefSSum_1): fresh64(X, X, Y, Z, W, V) = equiv2(Y, Z, V).
% 211.88/27.67 Axiom 11 (mDefSSum_1): fresh65(X, X, Y, Z, W, V) = fresh66(W, sdtpldt1(Y, Z), Y, Z, V).
% 211.88/27.67 Axiom 12 (mDefSSum_1): fresh63(X, X, Y, Z, W, V) = fresh64(aSet0(Y), true2, Y, Z, W, V).
% 211.88/27.67 Axiom 13 (mDefSSum_6): fresh(equiv2(X, Y, Z), true2, X, Y, Z) = sdtpldt0(w4(X, Y, Z), w5(X, Y, Z)).
% 211.88/27.67 Axiom 14 (mDefSSum_1): fresh63(aElementOf0(X, Y), true2, Z, W, Y, X) = fresh65(aSet0(W), true2, Z, W, Y, X).
% 211.88/27.67 Axiom 15 (mDefSSum_7): fresh21(equiv2(X, Y, Z), true2, X, Y, Z) = aElementOf0(w5(X, Y, Z), Y).
% 211.88/27.67 Axiom 16 (mDefSSum_8): fresh20(equiv2(X, Y, Z), true2, X, Y, Z) = aElementOf0(w4(X, Y, Z), X).
% 211.88/27.67
% 211.88/27.67 Lemma 17: equiv2(xI, xJ, xx) = true2.
% 211.88/27.67 Proof:
% 211.88/27.67 equiv2(xI, xJ, xx)
% 211.88/27.67 = { by axiom 10 (mDefSSum_1) R->L }
% 211.88/27.67 fresh64(true2, true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 211.88/27.67 = { by axiom 3 (mDefIdeal_7) R->L }
% 211.88/27.67 fresh64(fresh29(true2, true2, xI), true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 211.88/27.67 = { by axiom 1 (m__870) R->L }
% 211.88/27.67 fresh64(fresh29(aIdeal0(xI), true2, xI), true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 211.88/27.67 = { by axiom 5 (mDefIdeal_7) }
% 211.88/27.67 fresh64(aSet0(xI), true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 211.88/27.67 = { by axiom 12 (mDefSSum_1) R->L }
% 211.88/27.67 fresh63(true2, true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 211.88/27.67 = { by axiom 4 (m__901_1) R->L }
% 211.88/27.67 fresh63(aElementOf0(xx, sdtpldt1(xI, xJ)), true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 211.88/27.67 = { by axiom 14 (mDefSSum_1) }
% 212.70/27.67 fresh65(aSet0(xJ), true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 212.70/27.67 = { by axiom 5 (mDefIdeal_7) R->L }
% 212.70/27.67 fresh65(fresh29(aIdeal0(xJ), true2, xJ), true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 212.70/27.67 = { by axiom 2 (m__870_1) }
% 212.70/27.67 fresh65(fresh29(true2, true2, xJ), true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 212.70/27.67 = { by axiom 3 (mDefIdeal_7) }
% 212.70/27.67 fresh65(true2, true2, xI, xJ, sdtpldt1(xI, xJ), xx)
% 212.70/27.67 = { by axiom 11 (mDefSSum_1) }
% 212.70/27.67 fresh66(sdtpldt1(xI, xJ), sdtpldt1(xI, xJ), xI, xJ, xx)
% 212.70/27.67 = { by axiom 7 (mDefSSum_1) }
% 212.70/27.67 true2
% 212.70/27.67
% 212.70/27.67 Goal 1 (m__): tuple(xx, aElementOf0(X, xI), aElementOf0(Y, xJ)) = tuple(sdtpldt0(X, Y), true2, true2).
% 212.70/27.67 The goal is true when:
% 212.70/27.67 X = w4(xI, xJ, xx)
% 212.70/27.67 Y = w5(xI, xJ, xx)
% 212.70/27.67
% 212.70/27.67 Proof:
% 212.70/27.67 tuple(xx, aElementOf0(w4(xI, xJ, xx), xI), aElementOf0(w5(xI, xJ, xx), xJ))
% 212.70/27.67 = { by axiom 16 (mDefSSum_8) R->L }
% 212.70/27.67 tuple(xx, fresh20(equiv2(xI, xJ, xx), true2, xI, xJ, xx), aElementOf0(w5(xI, xJ, xx), xJ))
% 212.70/27.67 = { by lemma 17 }
% 212.70/27.67 tuple(xx, fresh20(true2, true2, xI, xJ, xx), aElementOf0(w5(xI, xJ, xx), xJ))
% 212.70/27.67 = { by axiom 9 (mDefSSum_8) }
% 212.70/27.67 tuple(xx, true2, aElementOf0(w5(xI, xJ, xx), xJ))
% 212.70/27.67 = { by axiom 15 (mDefSSum_7) R->L }
% 212.70/27.67 tuple(xx, true2, fresh21(equiv2(xI, xJ, xx), true2, xI, xJ, xx))
% 212.70/27.67 = { by lemma 17 }
% 212.70/27.67 tuple(xx, true2, fresh21(true2, true2, xI, xJ, xx))
% 212.70/27.67 = { by axiom 8 (mDefSSum_7) }
% 212.70/27.67 tuple(xx, true2, true2)
% 212.70/27.67 = { by axiom 6 (mDefSSum_6) R->L }
% 212.70/27.67 tuple(fresh(true2, true2, xI, xJ, xx), true2, true2)
% 212.70/27.67 = { by lemma 17 R->L }
% 212.70/27.67 tuple(fresh(equiv2(xI, xJ, xx), true2, xI, xJ, xx), true2, true2)
% 212.70/27.67 = { by axiom 13 (mDefSSum_6) }
% 212.70/27.67 tuple(sdtpldt0(w4(xI, xJ, xx), w5(xI, xJ, xx)), true2, true2)
% 212.70/27.67 % SZS output end Proof
% 212.70/27.67
% 212.70/27.67 RESULT: Theorem (the conjecture is true).
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