TSTP Solution File: COM013+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : COM013+4 : 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 : n022.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 : Wed Aug 30 18:45:21 EDT 2023
% Result : Theorem 1.77s 0.68s
% Output : Proof 2.75s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : COM013+4 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % 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 : n022.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 : Tue Aug 29 12:46:08 EDT 2023
% 0.13/0.35 % CPUTime :
% 1.77/0.68 Command-line arguments: --ground-connectedness --complete-subsets
% 1.77/0.68
% 1.77/0.68 % SZS status Theorem
% 1.77/0.68
% 2.75/0.72 % SZS output start Proof
% 2.75/0.72 Take the following subset of the input axioms:
% 2.75/0.73 fof(mNFRDef, definition, ![W0, W1]: ((aElement0(W0) & aRewritingSystem0(W1)) => ![W2]: (aNormalFormOfIn0(W2, W0, W1) <=> (aElement0(W2) & (sdtmndtasgtdt0(W0, W1, W2) & ~?[W3]: aReductOfIn0(W3, W2, W1)))))).
% 2.75/0.73 fof(mReduct, axiom, ![W0_2, W1_2]: ((aElement0(W0_2) & aRewritingSystem0(W1_2)) => ![W2_2]: (aReductOfIn0(W2_2, W0_2, W1_2) => aElement0(W2_2)))).
% 2.75/0.73 fof(mTCDef, definition, ![W2_2, W0_2, W1_2]: ((aElement0(W0_2) & (aRewritingSystem0(W1_2) & aElement0(W2_2))) => (sdtmndtplgtdt0(W0_2, W1_2, W2_2) <=> (aReductOfIn0(W2_2, W0_2, W1_2) | ?[W3_2]: (aElement0(W3_2) & (aReductOfIn0(W3_2, W0_2, W1_2) & sdtmndtplgtdt0(W3_2, W1_2, W2_2))))))).
% 2.75/0.73 fof(mTCRDef, definition, ![W2_2, W0_2, W1_2]: ((aElement0(W0_2) & (aRewritingSystem0(W1_2) & aElement0(W2_2))) => (sdtmndtasgtdt0(W0_2, W1_2, W2_2) <=> (W0_2=W2_2 | sdtmndtplgtdt0(W0_2, W1_2, W2_2))))).
% 2.75/0.73 fof(mTCRTrans, axiom, ![W2_2, W0_2, W1_2, W3_2]: ((aElement0(W0_2) & (aRewritingSystem0(W1_2) & (aElement0(W2_2) & aElement0(W3_2)))) => ((sdtmndtasgtdt0(W0_2, W1_2, W2_2) & sdtmndtasgtdt0(W2_2, W1_2, W3_2)) => sdtmndtasgtdt0(W0_2, W1_2, W3_2)))).
% 2.75/0.73 fof(m__, conjecture, ![W0_2]: (aElement0(W0_2) => (![W1_2]: (aElement0(W1_2) => (iLess0(W1_2, W0_2) => ?[W2_2]: (aElement0(W2_2) & ((W1_2=W2_2 | ((aReductOfIn0(W2_2, W1_2, xR) | ?[W3_2]: (aElement0(W3_2) & (aReductOfIn0(W3_2, W1_2, xR) & sdtmndtplgtdt0(W3_2, xR, W2_2)))) & sdtmndtplgtdt0(W1_2, xR, W2_2))) & (sdtmndtasgtdt0(W1_2, xR, W2_2) & (~?[W3_2]: aReductOfIn0(W3_2, W2_2, xR) & aNormalFormOfIn0(W2_2, W1_2, xR))))))) => ?[W1_2]: ((aElement0(W1_2) & ((W0_2=W1_2 | (aReductOfIn0(W1_2, W0_2, xR) | (?[W2_2]: (aElement0(W2_2) & (aReductOfIn0(W2_2, W0_2, xR) & sdtmndtplgtdt0(W2_2, xR, W1_2))) | (sdtmndtplgtdt0(W0_2, xR, W1_2) | sdtmndtasgtdt0(W0_2, xR, W1_2))))) & ~?[W2_2]: aReductOfIn0(W2_2, W1_2, xR))) | aNormalFormOfIn0(W1_2, W0_2, xR))))).
% 2.75/0.73 fof(m__587, hypothesis, aRewritingSystem0(xR) & (![W0_2, W1_2]: ((aElement0(W0_2) & aElement0(W1_2)) => ((aReductOfIn0(W1_2, W0_2, xR) | (?[W2_2]: (aElement0(W2_2) & (aReductOfIn0(W2_2, W0_2, xR) & sdtmndtplgtdt0(W2_2, xR, W1_2))) | sdtmndtplgtdt0(W0_2, xR, W1_2))) => iLess0(W1_2, W0_2))) & isTerminating0(xR))).
% 2.75/0.73
% 2.75/0.73 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.75/0.73 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.75/0.73 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.75/0.73 fresh(y, y, x1...xn) = u
% 2.75/0.73 C => fresh(s, t, x1...xn) = v
% 2.75/0.73 where fresh is a fresh function symbol and x1..xn are the free
% 2.75/0.73 variables of u and v.
% 2.75/0.73 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.75/0.73 input problem has no model of domain size 1).
% 2.75/0.73
% 2.75/0.73 The encoding turns the above axioms into the following unit equations and goals:
% 2.75/0.73
% 2.75/0.73 Axiom 1 (m__587): aRewritingSystem0(xR) = true2.
% 2.75/0.73 Axiom 2 (m__): aElement0(w0) = true2.
% 2.75/0.73 Axiom 3 (mReduct): fresh123(X, X, Y) = true2.
% 2.75/0.73 Axiom 4 (m___1): fresh27(X, X, Y) = true2.
% 2.75/0.73 Axiom 5 (m___1): fresh26(X, X, Y) = fresh27(w0, Y, Y).
% 2.75/0.73 Axiom 6 (m___13): fresh7(X, X, Y) = true2.
% 2.75/0.73 Axiom 7 (m___7): fresh6(X, X, Y) = aElement0(w2_2(Y)).
% 2.75/0.73 Axiom 8 (m___7): fresh5(X, X, Y) = true2.
% 2.75/0.73 Axiom 9 (m___8): fresh3(X, X, Y) = true2.
% 2.75/0.73 Axiom 10 (m__587_3): fresh34(X, X, Y, Z) = true2.
% 2.75/0.73 Axiom 11 (mReduct): fresh19(X, X, Y, Z) = aElement0(Z).
% 2.75/0.73 Axiom 12 (m__587_3): fresh14(X, X, Y, Z) = iLess0(Z, Y).
% 2.75/0.73 Axiom 13 (m___1): fresh26(aElement0(X), true2, X) = aReductOfIn0(w2(X), X, xR).
% 2.75/0.73 Axiom 14 (m___10): fresh23(X, X, Y) = aReductOfIn0(w2(Y), Y, xR).
% 2.75/0.73 Axiom 15 (m___13): fresh8(X, X, Y) = aReductOfIn0(w2(Y), Y, xR).
% 2.75/0.73 Axiom 16 (m___8): fresh4(X, X, Y) = sdtmndtasgtdt0(Y, xR, w2_2(Y)).
% 2.75/0.73 Axiom 17 (mReduct): fresh122(X, X, Y, Z, W) = fresh123(aElement0(Y), true2, W).
% 2.75/0.73 Axiom 18 (mTCDef_1): fresh121(X, X, Y, Z, W) = true2.
% 2.75/0.73 Axiom 19 (mTCDef_1): fresh119(X, X, Y, Z, W) = sdtmndtplgtdt0(Y, Z, W).
% 2.75/0.73 Axiom 20 (mTCRDef): fresh105(X, X, Y, Z, W) = true2.
% 2.75/0.73 Axiom 21 (mTCRDef): fresh103(X, X, Y, Z, W) = sdtmndtasgtdt0(Y, Z, W).
% 2.75/0.73 Axiom 22 (mTCRTrans): fresh101(X, X, Y, Z, W) = true2.
% 2.75/0.73 Axiom 23 (mTCRTrans): fresh99(X, X, Y, Z, W) = sdtmndtasgtdt0(Y, Z, W).
% 2.75/0.73 Axiom 24 (m__587_3): fresh33(X, X, Y, Z) = fresh34(aElement0(Y), true2, Y, Z).
% 2.75/0.73 Axiom 25 (m___7): fresh6(iLess0(X, w0), true2, X) = fresh5(aElement0(X), true2, X).
% 2.75/0.73 Axiom 26 (m___8): fresh4(iLess0(X, w0), true2, X) = fresh3(aElement0(X), true2, X).
% 2.75/0.73 Axiom 27 (mTCDef_1): fresh120(X, X, Y, Z, W) = fresh121(aElement0(Y), true2, Y, Z, W).
% 2.75/0.73 Axiom 28 (mTCDef_1): fresh118(X, X, Y, Z, W) = fresh119(aElement0(W), true2, Y, Z, W).
% 2.75/0.73 Axiom 29 (mTCRDef): fresh104(X, X, Y, Z, W) = fresh105(aElement0(Y), true2, Y, Z, W).
% 2.75/0.73 Axiom 30 (mTCRDef): fresh102(X, X, Y, Z, W) = fresh103(aElement0(W), true2, Y, Z, W).
% 2.75/0.73 Axiom 31 (mTCRTrans): fresh100(X, X, Y, Z, W, V) = fresh101(aElement0(Y), true2, Y, Z, V).
% 2.75/0.73 Axiom 32 (mTCRTrans): fresh98(X, X, Y, Z, W, V) = fresh99(aElement0(W), true2, Y, Z, V).
% 2.75/0.73 Axiom 33 (m___13): fresh8(sdtmndtasgtdt0(w0, xR, X), true2, X) = fresh7(aElement0(X), true2, X).
% 2.75/0.73 Axiom 34 (mTCRTrans): fresh97(X, X, Y, Z, W, V) = fresh100(aElement0(V), true2, Y, Z, W, V).
% 2.75/0.73 Axiom 35 (mTCRTrans): fresh96(X, X, Y, Z, W, V) = fresh98(aRewritingSystem0(Z), true2, Y, Z, W, V).
% 2.75/0.73 Axiom 36 (m__587_3): fresh33(aReductOfIn0(X, Y, xR), true2, Y, X) = fresh14(aElement0(X), true2, Y, X).
% 2.75/0.73 Axiom 37 (mReduct): fresh122(aReductOfIn0(X, Y, Z), true2, Y, Z, X) = fresh19(aRewritingSystem0(Z), true2, Y, X).
% 2.75/0.73 Axiom 38 (mTCDef_1): fresh118(aReductOfIn0(X, Y, Z), true2, Y, Z, X) = fresh120(aRewritingSystem0(Z), true2, Y, Z, X).
% 2.75/0.73 Axiom 39 (mTCRDef): fresh102(sdtmndtplgtdt0(X, Y, Z), true2, X, Y, Z) = fresh104(aRewritingSystem0(Y), true2, X, Y, Z).
% 2.75/0.73 Axiom 40 (mTCRTrans): fresh96(sdtmndtasgtdt0(X, Y, Z), true2, W, Y, X, Z) = fresh97(sdtmndtasgtdt0(W, Y, X), true2, W, Y, X, Z).
% 2.75/0.73
% 2.75/0.73 Lemma 41: fresh26(aElement0(X), true2, X) = fresh23(Y, Y, X).
% 2.75/0.73 Proof:
% 2.75/0.73 fresh26(aElement0(X), true2, X)
% 2.75/0.73 = { by axiom 13 (m___1) }
% 2.75/0.73 aReductOfIn0(w2(X), X, xR)
% 2.75/0.73 = { by axiom 14 (m___10) R->L }
% 2.75/0.73 fresh23(Y, Y, X)
% 2.75/0.73
% 2.75/0.73 Lemma 42: fresh23(X, X, w0) = true2.
% 2.75/0.73 Proof:
% 2.75/0.73 fresh23(X, X, w0)
% 2.75/0.73 = { by lemma 41 R->L }
% 2.75/0.73 fresh26(aElement0(w0), true2, w0)
% 2.75/0.73 = { by axiom 2 (m__) }
% 2.75/0.73 fresh26(true2, true2, w0)
% 2.75/0.73 = { by axiom 5 (m___1) }
% 2.75/0.73 fresh27(w0, w0, w0)
% 2.75/0.73 = { by axiom 4 (m___1) }
% 2.75/0.73 true2
% 2.75/0.73
% 2.75/0.73 Lemma 43: aElement0(w2(w0)) = true2.
% 2.75/0.73 Proof:
% 2.75/0.73 aElement0(w2(w0))
% 2.75/0.73 = { by axiom 11 (mReduct) R->L }
% 2.75/0.73 fresh19(true2, true2, w0, w2(w0))
% 2.75/0.73 = { by axiom 1 (m__587) R->L }
% 2.75/0.73 fresh19(aRewritingSystem0(xR), true2, w0, w2(w0))
% 2.75/0.73 = { by axiom 37 (mReduct) R->L }
% 2.75/0.73 fresh122(aReductOfIn0(w2(w0), w0, xR), true2, w0, xR, w2(w0))
% 2.75/0.73 = { by axiom 13 (m___1) R->L }
% 2.75/0.73 fresh122(fresh26(aElement0(w0), true2, w0), true2, w0, xR, w2(w0))
% 2.75/0.73 = { by lemma 41 }
% 2.75/0.73 fresh122(fresh23(X, X, w0), true2, w0, xR, w2(w0))
% 2.75/0.73 = { by lemma 42 }
% 2.75/0.73 fresh122(true2, true2, w0, xR, w2(w0))
% 2.75/0.73 = { by axiom 17 (mReduct) }
% 2.75/0.73 fresh123(aElement0(w0), true2, w2(w0))
% 2.75/0.73 = { by axiom 2 (m__) }
% 2.75/0.73 fresh123(true2, true2, w2(w0))
% 2.75/0.73 = { by axiom 3 (mReduct) }
% 2.75/0.73 true2
% 2.75/0.73
% 2.75/0.73 Lemma 44: iLess0(w2(w0), w0) = true2.
% 2.75/0.73 Proof:
% 2.75/0.73 iLess0(w2(w0), w0)
% 2.75/0.73 = { by axiom 12 (m__587_3) R->L }
% 2.75/0.73 fresh14(true2, true2, w0, w2(w0))
% 2.75/0.73 = { by lemma 43 R->L }
% 2.75/0.73 fresh14(aElement0(w2(w0)), true2, w0, w2(w0))
% 2.75/0.73 = { by axiom 36 (m__587_3) R->L }
% 2.75/0.73 fresh33(aReductOfIn0(w2(w0), w0, xR), true2, w0, w2(w0))
% 2.75/0.73 = { by axiom 13 (m___1) R->L }
% 2.75/0.73 fresh33(fresh26(aElement0(w0), true2, w0), true2, w0, w2(w0))
% 2.75/0.73 = { by lemma 41 }
% 2.75/0.73 fresh33(fresh23(X, X, w0), true2, w0, w2(w0))
% 2.75/0.73 = { by lemma 42 }
% 2.75/0.73 fresh33(true2, true2, w0, w2(w0))
% 2.75/0.73 = { by axiom 24 (m__587_3) }
% 2.75/0.73 fresh34(aElement0(w0), true2, w0, w2(w0))
% 2.75/0.73 = { by axiom 2 (m__) }
% 2.75/0.73 fresh34(true2, true2, w0, w2(w0))
% 2.75/0.73 = { by axiom 10 (m__587_3) }
% 2.75/0.73 true2
% 2.75/0.73
% 2.75/0.73 Lemma 45: aElement0(w2_2(w2(w0))) = true2.
% 2.75/0.73 Proof:
% 2.75/0.73 aElement0(w2_2(w2(w0)))
% 2.75/0.73 = { by axiom 7 (m___7) R->L }
% 2.75/0.73 fresh6(true2, true2, w2(w0))
% 2.75/0.73 = { by lemma 44 R->L }
% 2.75/0.73 fresh6(iLess0(w2(w0), w0), true2, w2(w0))
% 2.75/0.73 = { by axiom 25 (m___7) }
% 2.75/0.73 fresh5(aElement0(w2(w0)), true2, w2(w0))
% 2.75/0.73 = { by lemma 43 }
% 2.75/0.73 fresh5(true2, true2, w2(w0))
% 2.75/0.73 = { by axiom 8 (m___7) }
% 2.75/0.73 true2
% 2.75/0.73
% 2.75/0.73 Lemma 46: fresh23(X, X, w2_2(w2(w0))) = fresh26(Y, Y, w2_2(w2(w0))).
% 2.75/0.73 Proof:
% 2.75/0.73 fresh23(X, X, w2_2(w2(w0)))
% 2.75/0.73 = { by lemma 41 R->L }
% 2.75/0.73 fresh26(aElement0(w2_2(w2(w0))), true2, w2_2(w2(w0)))
% 2.75/0.73 = { by lemma 45 }
% 2.75/0.73 fresh26(true2, true2, w2_2(w2(w0)))
% 2.75/0.73 = { by axiom 5 (m___1) }
% 2.75/0.73 fresh27(w0, w2_2(w2(w0)), w2_2(w2(w0)))
% 2.75/0.73 = { by axiom 5 (m___1) R->L }
% 2.75/0.73 fresh26(Y, Y, w2_2(w2(w0)))
% 2.75/0.73
% 2.75/0.73 Goal 1 (m___2): tuple2(aElement0(X), aReductOfIn0(Y, w2_2(X), xR), iLess0(X, w0)) = tuple2(true2, true2, true2).
% 2.75/0.73 The goal is true when:
% 2.75/0.73 X = w2(w0)
% 2.75/0.73 Y = w2(w2_2(w2(w0)))
% 2.75/0.73
% 2.75/0.73 Proof:
% 2.75/0.73 tuple2(aElement0(w2(w0)), aReductOfIn0(w2(w2_2(w2(w0))), w2_2(w2(w0)), xR), iLess0(w2(w0), w0))
% 2.75/0.73 = { by axiom 13 (m___1) R->L }
% 2.75/0.73 tuple2(aElement0(w2(w0)), fresh26(aElement0(w2_2(w2(w0))), true2, w2_2(w2(w0))), iLess0(w2(w0), w0))
% 2.75/0.73 = { by lemma 41 }
% 2.75/0.73 tuple2(aElement0(w2(w0)), fresh23(W, W, w2_2(w2(w0))), iLess0(w2(w0), w0))
% 2.75/0.73 = { by lemma 43 }
% 2.75/0.73 tuple2(true2, fresh23(W, W, w2_2(w2(w0))), iLess0(w2(w0), w0))
% 2.75/0.73 = { by lemma 46 }
% 2.75/0.73 tuple2(true2, fresh26(Z, Z, w2_2(w2(w0))), iLess0(w2(w0), w0))
% 2.75/0.73 = { by lemma 44 }
% 2.75/0.73 tuple2(true2, fresh26(Z, Z, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 46 R->L }
% 2.75/0.73 tuple2(true2, fresh23(Y, Y, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 41 R->L }
% 2.75/0.73 tuple2(true2, fresh26(aElement0(w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 13 (m___1) }
% 2.75/0.73 tuple2(true2, aReductOfIn0(w2(w2_2(w2(w0))), w2_2(w2(w0)), xR), true2)
% 2.75/0.73 = { by axiom 15 (m___13) R->L }
% 2.75/0.73 tuple2(true2, fresh8(true2, true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 22 (mTCRTrans) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh101(true2, true2, w0, xR, w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 2 (m__) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh101(aElement0(w0), true2, w0, xR, w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 31 (mTCRTrans) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh100(true2, true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 45 R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh100(aElement0(w2_2(w2(w0))), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 34 (mTCRTrans) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(true2, true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 20 (mTCRDef) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh105(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 2 (m__) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh105(aElement0(w0), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 29 (mTCRDef) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh104(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 1 (m__587) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh104(aRewritingSystem0(xR), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 39 (mTCRDef) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(sdtmndtplgtdt0(w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 19 (mTCDef_1) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh119(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 43 R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh119(aElement0(w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 28 (mTCDef_1) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh118(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 42 R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh118(fresh23(X, X, w0), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 41 R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh118(fresh26(aElement0(w0), true2, w0), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 13 (m___1) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh118(aReductOfIn0(w2(w0), w0, xR), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 38 (mTCDef_1) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh120(aRewritingSystem0(xR), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 1 (m__587) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh120(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 27 (mTCDef_1) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh121(aElement0(w0), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 2 (m__) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(fresh121(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 18 (mTCDef_1) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh102(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 30 (mTCRDef) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh103(aElement0(w2(w0)), true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 43 }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(fresh103(true2, true2, w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 21 (mTCRDef) }
% 2.75/0.73 tuple2(true2, fresh8(fresh97(sdtmndtasgtdt0(w0, xR, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 40 (mTCRTrans) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh96(sdtmndtasgtdt0(w2(w0), xR, w2_2(w2(w0))), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 16 (m___8) R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh96(fresh4(true2, true2, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 44 R->L }
% 2.75/0.73 tuple2(true2, fresh8(fresh96(fresh4(iLess0(w2(w0), w0), true2, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 26 (m___8) }
% 2.75/0.73 tuple2(true2, fresh8(fresh96(fresh3(aElement0(w2(w0)), true2, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 43 }
% 2.75/0.73 tuple2(true2, fresh8(fresh96(fresh3(true2, true2, w2(w0)), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 9 (m___8) }
% 2.75/0.73 tuple2(true2, fresh8(fresh96(true2, true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 35 (mTCRTrans) }
% 2.75/0.73 tuple2(true2, fresh8(fresh98(aRewritingSystem0(xR), true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 1 (m__587) }
% 2.75/0.73 tuple2(true2, fresh8(fresh98(true2, true2, w0, xR, w2(w0), w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 32 (mTCRTrans) }
% 2.75/0.73 tuple2(true2, fresh8(fresh99(aElement0(w2(w0)), true2, w0, xR, w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 43 }
% 2.75/0.73 tuple2(true2, fresh8(fresh99(true2, true2, w0, xR, w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 23 (mTCRTrans) }
% 2.75/0.73 tuple2(true2, fresh8(sdtmndtasgtdt0(w0, xR, w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 33 (m___13) }
% 2.75/0.73 tuple2(true2, fresh7(aElement0(w2_2(w2(w0))), true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by lemma 45 }
% 2.75/0.73 tuple2(true2, fresh7(true2, true2, w2_2(w2(w0))), true2)
% 2.75/0.73 = { by axiom 6 (m___13) }
% 2.75/0.73 tuple2(true2, true2, true2)
% 2.75/0.73 % SZS output end Proof
% 2.75/0.73
% 2.75/0.73 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------