TSTP Solution File: NUM422+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM422+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 : 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 11:56:12 EDT 2023
% Result : Theorem 0.20s 0.70s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : NUM422+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
% 0.13/0.35 % Computer : n011.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 15:22:08 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.70 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.20/0.70
% 0.20/0.70 % SZS status Theorem
% 0.20/0.70
% 0.20/0.71 % SZS output start Proof
% 0.20/0.71 Take the following subset of the input axioms:
% 0.20/0.71 fof(mAddAsso, axiom, ![W0, W1, W2]: ((aInteger0(W0) & (aInteger0(W1) & aInteger0(W2))) => sdtpldt0(W0, sdtpldt0(W1, W2))=sdtpldt0(sdtpldt0(W0, W1), W2))).
% 0.20/0.71 fof(mAddNeg, axiom, ![W0_2]: (aInteger0(W0_2) => (sdtpldt0(W0_2, smndt0(W0_2))=sz00 & sz00=sdtpldt0(smndt0(W0_2), W0_2)))).
% 0.20/0.71 fof(mAddZero, axiom, ![W0_2]: (aInteger0(W0_2) => (sdtpldt0(W0_2, sz00)=W0_2 & W0_2=sdtpldt0(sz00, W0_2)))).
% 0.20/0.71 fof(mDistrib, axiom, ![W0_2, W1_2, W2_2]: ((aInteger0(W0_2) & (aInteger0(W1_2) & aInteger0(W2_2))) => (sdtasdt0(W0_2, sdtpldt0(W1_2, W2_2))=sdtpldt0(sdtasdt0(W0_2, W1_2), sdtasdt0(W0_2, W2_2)) & sdtasdt0(sdtpldt0(W0_2, W1_2), W2_2)=sdtpldt0(sdtasdt0(W0_2, W2_2), sdtasdt0(W1_2, W2_2))))).
% 0.20/0.71 fof(mIntMult, axiom, ![W0_2, W1_2]: ((aInteger0(W0_2) & aInteger0(W1_2)) => aInteger0(sdtasdt0(W0_2, W1_2)))).
% 0.20/0.71 fof(mIntNeg, axiom, ![W0_2]: (aInteger0(W0_2) => aInteger0(smndt0(W0_2)))).
% 0.20/0.71 fof(mIntOne, axiom, aInteger0(sz10)).
% 0.20/0.71 fof(mMulComm, axiom, ![W0_2, W1_2]: ((aInteger0(W0_2) & aInteger0(W1_2)) => sdtasdt0(W0_2, W1_2)=sdtasdt0(W1_2, W0_2))).
% 0.20/0.71 fof(mMulOne, axiom, ![W0_2]: (aInteger0(W0_2) => (sdtasdt0(W0_2, sz10)=W0_2 & W0_2=sdtasdt0(sz10, W0_2)))).
% 0.20/0.71 fof(mMulZero, axiom, ![W0_2]: (aInteger0(W0_2) => (sdtasdt0(W0_2, sz00)=sz00 & sz00=sdtasdt0(sz00, W0_2)))).
% 0.20/0.71 fof(m__, conjecture, sdtasdt0(smndt0(sz10), xa)=smndt0(xa)).
% 0.20/0.71 fof(m__446, hypothesis, aInteger0(xa)).
% 0.20/0.71
% 0.20/0.71 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.71 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.71 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.71 fresh(y, y, x1...xn) = u
% 0.20/0.71 C => fresh(s, t, x1...xn) = v
% 0.20/0.71 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.71 variables of u and v.
% 0.20/0.71 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.71 input problem has no model of domain size 1).
% 0.20/0.71
% 0.20/0.71 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.71
% 0.20/0.71 Axiom 1 (mIntOne): aInteger0(sz10) = true.
% 0.20/0.71 Axiom 2 (m__446): aInteger0(xa) = true.
% 0.20/0.71 Axiom 3 (mAddNeg): fresh18(X, X, Y) = sz00.
% 0.20/0.71 Axiom 4 (mAddNeg_1): fresh17(X, X, Y) = sz00.
% 0.20/0.71 Axiom 5 (mIntNeg): fresh12(X, X, Y) = true.
% 0.20/0.71 Axiom 6 (mMulZero_1): fresh5(X, X, Y) = sz00.
% 0.20/0.71 Axiom 7 (mAddZero_1): fresh4(X, X, Y) = Y.
% 0.20/0.71 Axiom 8 (mAddZero): fresh3(X, X, Y) = Y.
% 0.20/0.71 Axiom 9 (mMulOne_1): fresh2(X, X, Y) = Y.
% 0.20/0.71 Axiom 10 (mAddNeg): fresh18(aInteger0(X), true, X) = sdtpldt0(X, smndt0(X)).
% 0.20/0.71 Axiom 11 (mAddNeg_1): fresh17(aInteger0(X), true, X) = sdtpldt0(smndt0(X), X).
% 0.20/0.71 Axiom 12 (mIntMult): fresh14(X, X, Y, Z) = aInteger0(sdtasdt0(Y, Z)).
% 0.20/0.71 Axiom 13 (mIntMult): fresh13(X, X, Y, Z) = true.
% 0.20/0.71 Axiom 14 (mIntNeg): fresh12(aInteger0(X), true, X) = aInteger0(smndt0(X)).
% 0.20/0.71 Axiom 15 (mMulComm): fresh8(X, X, Y, Z) = sdtasdt0(Y, Z).
% 0.20/0.71 Axiom 16 (mMulComm): fresh7(X, X, Y, Z) = sdtasdt0(Z, Y).
% 0.20/0.71 Axiom 17 (mMulZero_1): fresh5(aInteger0(X), true, X) = sdtasdt0(sz00, X).
% 0.20/0.71 Axiom 18 (mAddZero_1): fresh4(aInteger0(X), true, X) = sdtpldt0(sz00, X).
% 0.20/0.71 Axiom 19 (mAddZero): fresh3(aInteger0(X), true, X) = sdtpldt0(X, sz00).
% 0.20/0.71 Axiom 20 (mMulOne_1): fresh2(aInteger0(X), true, X) = sdtasdt0(sz10, X).
% 0.20/0.71 Axiom 21 (mAddAsso): fresh29(X, X, Y, Z, W) = sdtpldt0(sdtpldt0(Y, Z), W).
% 0.20/0.71 Axiom 22 (mDistrib_1): fresh25(X, X, Y, Z, W) = sdtasdt0(sdtpldt0(Y, Z), W).
% 0.20/0.71 Axiom 23 (mAddAsso): fresh21(X, X, Y, Z, W) = sdtpldt0(Y, sdtpldt0(Z, W)).
% 0.20/0.71 Axiom 24 (mIntMult): fresh14(aInteger0(X), true, Y, X) = fresh13(aInteger0(Y), true, Y, X).
% 0.20/0.71 Axiom 25 (mMulComm): fresh8(aInteger0(X), true, Y, X) = fresh7(aInteger0(Y), true, Y, X).
% 0.20/0.71 Axiom 26 (mDistrib_1): fresh15(X, X, Y, Z, W) = sdtpldt0(sdtasdt0(Y, W), sdtasdt0(Z, W)).
% 0.20/0.71 Axiom 27 (mAddAsso): fresh28(X, X, Y, Z, W) = fresh29(aInteger0(Y), true, Y, Z, W).
% 0.20/0.71 Axiom 28 (mDistrib_1): fresh24(X, X, Y, Z, W) = fresh25(aInteger0(Y), true, Y, Z, W).
% 0.20/0.71 Axiom 29 (mAddAsso): fresh28(aInteger0(X), true, Y, Z, X) = fresh21(aInteger0(Z), true, Y, Z, X).
% 0.20/0.71 Axiom 30 (mDistrib_1): fresh24(aInteger0(X), true, Y, Z, X) = fresh15(aInteger0(Z), true, Y, Z, X).
% 0.20/0.71
% 0.20/0.71 Lemma 31: aInteger0(smndt0(sz10)) = true.
% 0.20/0.71 Proof:
% 0.20/0.71 aInteger0(smndt0(sz10))
% 0.20/0.72 = { by axiom 14 (mIntNeg) R->L }
% 0.20/0.72 fresh12(aInteger0(sz10), true, sz10)
% 0.20/0.72 = { by axiom 1 (mIntOne) }
% 0.20/0.72 fresh12(true, true, sz10)
% 0.20/0.72 = { by axiom 5 (mIntNeg) }
% 0.20/0.72 true
% 0.20/0.72
% 0.20/0.72 Lemma 32: aInteger0(smndt0(xa)) = true.
% 0.20/0.72 Proof:
% 0.20/0.72 aInteger0(smndt0(xa))
% 0.20/0.72 = { by axiom 14 (mIntNeg) R->L }
% 0.20/0.72 fresh12(aInteger0(xa), true, xa)
% 0.20/0.72 = { by axiom 2 (m__446) }
% 0.20/0.72 fresh12(true, true, xa)
% 0.20/0.72 = { by axiom 5 (mIntNeg) }
% 0.20/0.72 true
% 0.20/0.72
% 0.20/0.72 Lemma 33: aInteger0(sdtasdt0(smndt0(sz10), xa)) = true.
% 0.20/0.72 Proof:
% 0.20/0.72 aInteger0(sdtasdt0(smndt0(sz10), xa))
% 0.20/0.72 = { by axiom 16 (mMulComm) R->L }
% 0.20/0.72 aInteger0(fresh7(true, true, xa, smndt0(sz10)))
% 0.20/0.72 = { by axiom 2 (m__446) R->L }
% 0.20/0.72 aInteger0(fresh7(aInteger0(xa), true, xa, smndt0(sz10)))
% 0.20/0.72 = { by axiom 25 (mMulComm) R->L }
% 0.20/0.72 aInteger0(fresh8(aInteger0(smndt0(sz10)), true, xa, smndt0(sz10)))
% 0.20/0.72 = { by lemma 31 }
% 0.20/0.72 aInteger0(fresh8(true, true, xa, smndt0(sz10)))
% 0.20/0.72 = { by axiom 15 (mMulComm) }
% 0.20/0.72 aInteger0(sdtasdt0(xa, smndt0(sz10)))
% 0.20/0.72 = { by axiom 12 (mIntMult) R->L }
% 0.20/0.72 fresh14(true, true, xa, smndt0(sz10))
% 0.20/0.72 = { by lemma 31 R->L }
% 0.20/0.72 fresh14(aInteger0(smndt0(sz10)), true, xa, smndt0(sz10))
% 0.20/0.72 = { by axiom 24 (mIntMult) }
% 0.20/0.72 fresh13(aInteger0(xa), true, xa, smndt0(sz10))
% 0.20/0.72 = { by axiom 2 (m__446) }
% 0.20/0.72 fresh13(true, true, xa, smndt0(sz10))
% 0.20/0.72 = { by axiom 13 (mIntMult) }
% 0.20/0.72 true
% 0.20/0.72
% 0.20/0.72 Goal 1 (m__): sdtasdt0(smndt0(sz10), xa) = smndt0(xa).
% 0.20/0.72 Proof:
% 0.20/0.72 sdtasdt0(smndt0(sz10), xa)
% 0.20/0.72 = { by axiom 8 (mAddZero) R->L }
% 0.20/0.72 fresh3(true, true, sdtasdt0(smndt0(sz10), xa))
% 0.20/0.72 = { by lemma 33 R->L }
% 0.20/0.72 fresh3(aInteger0(sdtasdt0(smndt0(sz10), xa)), true, sdtasdt0(smndt0(sz10), xa))
% 0.20/0.72 = { by axiom 19 (mAddZero) }
% 0.20/0.72 sdtpldt0(sdtasdt0(smndt0(sz10), xa), sz00)
% 0.20/0.72 = { by axiom 3 (mAddNeg) R->L }
% 0.20/0.72 sdtpldt0(sdtasdt0(smndt0(sz10), xa), fresh18(true, true, xa))
% 0.20/0.72 = { by axiom 2 (m__446) R->L }
% 0.20/0.72 sdtpldt0(sdtasdt0(smndt0(sz10), xa), fresh18(aInteger0(xa), true, xa))
% 0.20/0.72 = { by axiom 10 (mAddNeg) }
% 0.20/0.72 sdtpldt0(sdtasdt0(smndt0(sz10), xa), sdtpldt0(xa, smndt0(xa)))
% 0.20/0.72 = { by axiom 23 (mAddAsso) R->L }
% 0.20/0.72 fresh21(true, true, sdtasdt0(smndt0(sz10), xa), xa, smndt0(xa))
% 0.20/0.72 = { by axiom 2 (m__446) R->L }
% 0.20/0.72 fresh21(aInteger0(xa), true, sdtasdt0(smndt0(sz10), xa), xa, smndt0(xa))
% 0.20/0.72 = { by axiom 29 (mAddAsso) R->L }
% 0.20/0.72 fresh28(aInteger0(smndt0(xa)), true, sdtasdt0(smndt0(sz10), xa), xa, smndt0(xa))
% 0.20/0.72 = { by lemma 32 }
% 0.20/0.72 fresh28(true, true, sdtasdt0(smndt0(sz10), xa), xa, smndt0(xa))
% 0.20/0.72 = { by axiom 27 (mAddAsso) }
% 0.20/0.72 fresh29(aInteger0(sdtasdt0(smndt0(sz10), xa)), true, sdtasdt0(smndt0(sz10), xa), xa, smndt0(xa))
% 0.20/0.72 = { by lemma 33 }
% 0.20/0.72 fresh29(true, true, sdtasdt0(smndt0(sz10), xa), xa, smndt0(xa))
% 0.20/0.72 = { by axiom 21 (mAddAsso) }
% 0.20/0.72 sdtpldt0(sdtpldt0(sdtasdt0(smndt0(sz10), xa), xa), smndt0(xa))
% 0.20/0.72 = { by axiom 9 (mMulOne_1) R->L }
% 0.20/0.72 sdtpldt0(sdtpldt0(sdtasdt0(smndt0(sz10), xa), fresh2(true, true, xa)), smndt0(xa))
% 0.20/0.72 = { by axiom 2 (m__446) R->L }
% 0.20/0.72 sdtpldt0(sdtpldt0(sdtasdt0(smndt0(sz10), xa), fresh2(aInteger0(xa), true, xa)), smndt0(xa))
% 0.20/0.72 = { by axiom 20 (mMulOne_1) }
% 0.20/0.72 sdtpldt0(sdtpldt0(sdtasdt0(smndt0(sz10), xa), sdtasdt0(sz10, xa)), smndt0(xa))
% 0.20/0.72 = { by axiom 26 (mDistrib_1) R->L }
% 0.20/0.72 sdtpldt0(fresh15(true, true, smndt0(sz10), sz10, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 1 (mIntOne) R->L }
% 0.20/0.72 sdtpldt0(fresh15(aInteger0(sz10), true, smndt0(sz10), sz10, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 30 (mDistrib_1) R->L }
% 0.20/0.72 sdtpldt0(fresh24(aInteger0(xa), true, smndt0(sz10), sz10, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 2 (m__446) }
% 0.20/0.72 sdtpldt0(fresh24(true, true, smndt0(sz10), sz10, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 28 (mDistrib_1) }
% 0.20/0.72 sdtpldt0(fresh25(aInteger0(smndt0(sz10)), true, smndt0(sz10), sz10, xa), smndt0(xa))
% 0.20/0.72 = { by lemma 31 }
% 0.20/0.72 sdtpldt0(fresh25(true, true, smndt0(sz10), sz10, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 22 (mDistrib_1) }
% 0.20/0.72 sdtpldt0(sdtasdt0(sdtpldt0(smndt0(sz10), sz10), xa), smndt0(xa))
% 0.20/0.72 = { by axiom 11 (mAddNeg_1) R->L }
% 0.20/0.72 sdtpldt0(sdtasdt0(fresh17(aInteger0(sz10), true, sz10), xa), smndt0(xa))
% 0.20/0.72 = { by axiom 1 (mIntOne) }
% 0.20/0.72 sdtpldt0(sdtasdt0(fresh17(true, true, sz10), xa), smndt0(xa))
% 0.20/0.72 = { by axiom 4 (mAddNeg_1) }
% 0.20/0.72 sdtpldt0(sdtasdt0(sz00, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 17 (mMulZero_1) R->L }
% 0.20/0.72 sdtpldt0(fresh5(aInteger0(xa), true, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 2 (m__446) }
% 0.20/0.72 sdtpldt0(fresh5(true, true, xa), smndt0(xa))
% 0.20/0.72 = { by axiom 6 (mMulZero_1) }
% 0.20/0.72 sdtpldt0(sz00, smndt0(xa))
% 0.20/0.72 = { by axiom 18 (mAddZero_1) R->L }
% 0.20/0.72 fresh4(aInteger0(smndt0(xa)), true, smndt0(xa))
% 0.20/0.72 = { by lemma 32 }
% 0.20/0.72 fresh4(true, true, smndt0(xa))
% 0.20/0.72 = { by axiom 7 (mAddZero_1) }
% 0.20/0.72 smndt0(xa)
% 0.20/0.72 % SZS output end Proof
% 0.20/0.72
% 0.20/0.72 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------