TSTP Solution File: NUM421+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM421+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 : n010.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 0.21s 0.71s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : NUM421+1 : TPTP v8.1.2. Released v4.0.0.
% 0.08/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n010.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Fri Aug 25 17:20:20 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.21/0.71 Command-line arguments: --ground-connectedness --complete-subsets
% 0.21/0.71
% 0.21/0.71 % SZS status Theorem
% 0.21/0.72
% 0.21/0.73 % SZS output start Proof
% 0.21/0.73 Take the following subset of the input axioms:
% 0.21/0.73 fof(mAddAsso, axiom, ![W0, W1, W2]: ((aInteger0(W0) & (aInteger0(W1) & aInteger0(W2))) => sdtpldt0(W0, sdtpldt0(W1, W2))=sdtpldt0(sdtpldt0(W0, W1), W2))).
% 0.21/0.73 fof(mAddComm, axiom, ![W0_2, W1_2]: ((aInteger0(W0_2) & aInteger0(W1_2)) => sdtpldt0(W0_2, W1_2)=sdtpldt0(W1_2, W0_2))).
% 0.21/0.73 fof(mAddNeg, axiom, ![W0_2]: (aInteger0(W0_2) => (sdtpldt0(W0_2, smndt0(W0_2))=sz00 & sz00=sdtpldt0(smndt0(W0_2), W0_2)))).
% 0.21/0.73 fof(mAddZero, axiom, ![W0_2]: (aInteger0(W0_2) => (sdtpldt0(W0_2, sz00)=W0_2 & W0_2=sdtpldt0(sz00, W0_2)))).
% 0.21/0.73 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))))).
% 0.21/0.73 fof(mIntMult, axiom, ![W0_2, W1_2]: ((aInteger0(W0_2) & aInteger0(W1_2)) => aInteger0(sdtasdt0(W0_2, W1_2)))).
% 0.21/0.73 fof(mIntNeg, axiom, ![W0_2]: (aInteger0(W0_2) => aInteger0(smndt0(W0_2)))).
% 0.21/0.73 fof(mIntOne, axiom, aInteger0(sz10)).
% 0.21/0.73 fof(mIntZero, axiom, aInteger0(sz00)).
% 0.21/0.73 fof(mMulComm, axiom, ![W0_2, W1_2]: ((aInteger0(W0_2) & aInteger0(W1_2)) => sdtasdt0(W0_2, W1_2)=sdtasdt0(W1_2, W0_2))).
% 0.21/0.73 fof(mMulOne, axiom, ![W0_2]: (aInteger0(W0_2) => (sdtasdt0(W0_2, sz10)=W0_2 & W0_2=sdtasdt0(sz10, W0_2)))).
% 0.21/0.73 fof(m__, conjecture, sdtasdt0(xa, sz00)=sz00 & sz00=sdtasdt0(sz00, xa)).
% 0.21/0.73 fof(m__419, hypothesis, aInteger0(xa)).
% 0.21/0.73
% 0.21/0.73 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.73 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.73 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.73 fresh(y, y, x1...xn) = u
% 0.21/0.73 C => fresh(s, t, x1...xn) = v
% 0.21/0.73 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.73 variables of u and v.
% 0.21/0.73 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.73 input problem has no model of domain size 1).
% 0.21/0.73
% 0.21/0.73 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.73
% 0.21/0.74 Axiom 1 (mIntOne): aInteger0(sz10) = true.
% 0.21/0.74 Axiom 2 (m__419): aInteger0(xa) = true.
% 0.21/0.74 Axiom 3 (mIntZero): aInteger0(sz00) = true.
% 0.21/0.74 Axiom 4 (mMulOne): fresh(X, X, Y) = Y.
% 0.21/0.74 Axiom 5 (mAddNeg): fresh16(X, X, Y) = sz00.
% 0.21/0.74 Axiom 6 (mIntNeg): fresh10(X, X, Y) = true.
% 0.21/0.74 Axiom 7 (mAddZero): fresh3(X, X, Y) = Y.
% 0.21/0.74 Axiom 8 (mMulOne): fresh(aInteger0(X), true, X) = sdtasdt0(X, sz10).
% 0.21/0.74 Axiom 9 (mAddComm): fresh18(X, X, Y, Z) = sdtpldt0(Y, Z).
% 0.21/0.74 Axiom 10 (mAddComm): fresh17(X, X, Y, Z) = sdtpldt0(Z, Y).
% 0.21/0.74 Axiom 11 (mAddNeg): fresh16(aInteger0(X), true, X) = sdtpldt0(X, smndt0(X)).
% 0.21/0.74 Axiom 12 (mIntMult): fresh12(X, X, Y, Z) = aInteger0(sdtasdt0(Y, Z)).
% 0.21/0.74 Axiom 13 (mIntMult): fresh11(X, X, Y, Z) = true.
% 0.21/0.74 Axiom 14 (mIntNeg): fresh10(aInteger0(X), true, X) = aInteger0(smndt0(X)).
% 0.21/0.74 Axiom 15 (mMulComm): fresh6(X, X, Y, Z) = sdtasdt0(Y, Z).
% 0.21/0.74 Axiom 16 (mMulComm): fresh5(X, X, Y, Z) = sdtasdt0(Z, Y).
% 0.21/0.74 Axiom 17 (mAddZero): fresh3(aInteger0(X), true, X) = sdtpldt0(X, sz00).
% 0.21/0.74 Axiom 18 (mAddAsso): fresh27(X, X, Y, Z, W) = sdtpldt0(sdtpldt0(Y, Z), W).
% 0.21/0.74 Axiom 19 (mDistrib): fresh21(X, X, Y, Z, W) = sdtasdt0(Y, sdtpldt0(Z, W)).
% 0.21/0.74 Axiom 20 (mAddAsso): fresh19(X, X, Y, Z, W) = sdtpldt0(Y, sdtpldt0(Z, W)).
% 0.21/0.74 Axiom 21 (mAddComm): fresh18(aInteger0(X), true, Y, X) = fresh17(aInteger0(Y), true, Y, X).
% 0.21/0.74 Axiom 22 (mIntMult): fresh12(aInteger0(X), true, Y, X) = fresh11(aInteger0(Y), true, Y, X).
% 0.21/0.74 Axiom 23 (mMulComm): fresh6(aInteger0(X), true, Y, X) = fresh5(aInteger0(Y), true, Y, X).
% 0.21/0.74 Axiom 24 (mAddAsso): fresh26(X, X, Y, Z, W) = fresh27(aInteger0(Y), true, Y, Z, W).
% 0.21/0.74 Axiom 25 (mDistrib): fresh20(X, X, Y, Z, W) = fresh21(aInteger0(Y), true, Y, Z, W).
% 0.21/0.74 Axiom 26 (mAddAsso): fresh26(aInteger0(X), true, Y, Z, X) = fresh19(aInteger0(Z), true, Y, Z, X).
% 0.21/0.74 Axiom 27 (mDistrib): fresh20(aInteger0(X), true, Y, Z, X) = fresh14(aInteger0(Z), true, Y, Z, X).
% 0.21/0.74 Axiom 28 (mDistrib): fresh14(X, X, Y, Z, W) = sdtpldt0(sdtasdt0(Y, Z), sdtasdt0(Y, W)).
% 0.21/0.74
% 0.21/0.74 Lemma 29: aInteger0(sdtasdt0(sz00, xa)) = true.
% 0.21/0.74 Proof:
% 0.21/0.74 aInteger0(sdtasdt0(sz00, xa))
% 0.21/0.74 = { by axiom 12 (mIntMult) R->L }
% 0.21/0.74 fresh12(true, true, sz00, xa)
% 0.21/0.74 = { by axiom 2 (m__419) R->L }
% 0.21/0.74 fresh12(aInteger0(xa), true, sz00, xa)
% 0.21/0.74 = { by axiom 22 (mIntMult) }
% 0.21/0.74 fresh11(aInteger0(sz00), true, sz00, xa)
% 0.21/0.74 = { by axiom 3 (mIntZero) }
% 0.21/0.74 fresh11(true, true, sz00, xa)
% 0.21/0.74 = { by axiom 13 (mIntMult) }
% 0.21/0.74 true
% 0.21/0.74
% 0.21/0.74 Lemma 30: sdtpldt0(xa, smndt0(xa)) = sz00.
% 0.21/0.74 Proof:
% 0.21/0.74 sdtpldt0(xa, smndt0(xa))
% 0.21/0.74 = { by axiom 11 (mAddNeg) R->L }
% 0.21/0.74 fresh16(aInteger0(xa), true, xa)
% 0.21/0.74 = { by axiom 2 (m__419) }
% 0.21/0.74 fresh16(true, true, xa)
% 0.21/0.74 = { by axiom 5 (mAddNeg) }
% 0.21/0.74 sz00
% 0.21/0.74
% 0.21/0.74 Lemma 31: sdtasdt0(sz00, xa) = sdtasdt0(xa, sz00).
% 0.21/0.74 Proof:
% 0.21/0.74 sdtasdt0(sz00, xa)
% 0.21/0.74 = { by axiom 16 (mMulComm) R->L }
% 0.21/0.74 fresh5(true, true, xa, sz00)
% 0.21/0.74 = { by axiom 2 (m__419) R->L }
% 0.21/0.74 fresh5(aInteger0(xa), true, xa, sz00)
% 0.21/0.74 = { by axiom 23 (mMulComm) R->L }
% 0.21/0.74 fresh6(aInteger0(sz00), true, xa, sz00)
% 0.21/0.74 = { by axiom 3 (mIntZero) }
% 0.21/0.74 fresh6(true, true, xa, sz00)
% 0.21/0.74 = { by axiom 15 (mMulComm) }
% 0.21/0.74 sdtasdt0(xa, sz00)
% 0.21/0.74
% 0.21/0.74 Lemma 32: sdtasdt0(xa, sz10) = xa.
% 0.21/0.74 Proof:
% 0.21/0.74 sdtasdt0(xa, sz10)
% 0.21/0.74 = { by axiom 8 (mMulOne) R->L }
% 0.21/0.74 fresh(aInteger0(xa), true, xa)
% 0.21/0.74 = { by axiom 2 (m__419) }
% 0.21/0.74 fresh(true, true, xa)
% 0.21/0.74 = { by axiom 4 (mMulOne) }
% 0.21/0.74 xa
% 0.21/0.74
% 0.21/0.74 Lemma 33: sdtasdt0(sz00, xa) = sz00.
% 0.21/0.74 Proof:
% 0.21/0.74 sdtasdt0(sz00, xa)
% 0.21/0.74 = { by axiom 7 (mAddZero) R->L }
% 0.21/0.74 fresh3(true, true, sdtasdt0(sz00, xa))
% 0.21/0.74 = { by lemma 29 R->L }
% 0.21/0.74 fresh3(aInteger0(sdtasdt0(sz00, xa)), true, sdtasdt0(sz00, xa))
% 0.21/0.74 = { by axiom 17 (mAddZero) }
% 0.21/0.74 sdtpldt0(sdtasdt0(sz00, xa), sz00)
% 0.21/0.74 = { by lemma 30 R->L }
% 0.21/0.74 sdtpldt0(sdtasdt0(sz00, xa), sdtpldt0(xa, smndt0(xa)))
% 0.21/0.74 = { by axiom 20 (mAddAsso) R->L }
% 0.21/0.74 fresh19(true, true, sdtasdt0(sz00, xa), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 2 (m__419) R->L }
% 0.21/0.74 fresh19(aInteger0(xa), true, sdtasdt0(sz00, xa), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 26 (mAddAsso) R->L }
% 0.21/0.74 fresh26(aInteger0(smndt0(xa)), true, sdtasdt0(sz00, xa), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 14 (mIntNeg) R->L }
% 0.21/0.74 fresh26(fresh10(aInteger0(xa), true, xa), true, sdtasdt0(sz00, xa), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 2 (m__419) }
% 0.21/0.74 fresh26(fresh10(true, true, xa), true, sdtasdt0(sz00, xa), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 6 (mIntNeg) }
% 0.21/0.74 fresh26(true, true, sdtasdt0(sz00, xa), xa, smndt0(xa))
% 0.21/0.74 = { by lemma 31 }
% 0.21/0.74 fresh26(true, true, sdtasdt0(xa, sz00), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 24 (mAddAsso) }
% 0.21/0.74 fresh27(aInteger0(sdtasdt0(xa, sz00)), true, sdtasdt0(xa, sz00), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 12 (mIntMult) R->L }
% 0.21/0.74 fresh27(fresh12(true, true, xa, sz00), true, sdtasdt0(xa, sz00), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 3 (mIntZero) R->L }
% 0.21/0.74 fresh27(fresh12(aInteger0(sz00), true, xa, sz00), true, sdtasdt0(xa, sz00), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 22 (mIntMult) }
% 0.21/0.74 fresh27(fresh11(aInteger0(xa), true, xa, sz00), true, sdtasdt0(xa, sz00), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 2 (m__419) }
% 0.21/0.74 fresh27(fresh11(true, true, xa, sz00), true, sdtasdt0(xa, sz00), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 13 (mIntMult) }
% 0.21/0.74 fresh27(true, true, sdtasdt0(xa, sz00), xa, smndt0(xa))
% 0.21/0.74 = { by axiom 18 (mAddAsso) }
% 0.21/0.74 sdtpldt0(sdtpldt0(sdtasdt0(xa, sz00), xa), smndt0(xa))
% 0.21/0.74 = { by lemma 31 R->L }
% 0.21/0.74 sdtpldt0(sdtpldt0(sdtasdt0(sz00, xa), xa), smndt0(xa))
% 0.21/0.74 = { by axiom 10 (mAddComm) R->L }
% 0.21/0.74 sdtpldt0(fresh17(true, true, xa, sdtasdt0(sz00, xa)), smndt0(xa))
% 0.21/0.74 = { by axiom 2 (m__419) R->L }
% 0.21/0.74 sdtpldt0(fresh17(aInteger0(xa), true, xa, sdtasdt0(sz00, xa)), smndt0(xa))
% 0.21/0.74 = { by axiom 21 (mAddComm) R->L }
% 0.21/0.74 sdtpldt0(fresh18(aInteger0(sdtasdt0(sz00, xa)), true, xa, sdtasdt0(sz00, xa)), smndt0(xa))
% 0.21/0.74 = { by lemma 29 }
% 0.21/0.74 sdtpldt0(fresh18(true, true, xa, sdtasdt0(sz00, xa)), smndt0(xa))
% 0.21/0.74 = { by axiom 9 (mAddComm) }
% 0.21/0.74 sdtpldt0(sdtpldt0(xa, sdtasdt0(sz00, xa)), smndt0(xa))
% 0.21/0.74 = { by lemma 32 R->L }
% 0.21/0.74 sdtpldt0(sdtpldt0(sdtasdt0(xa, sz10), sdtasdt0(sz00, xa)), smndt0(xa))
% 0.21/0.74 = { by lemma 31 }
% 0.21/0.74 sdtpldt0(sdtpldt0(sdtasdt0(xa, sz10), sdtasdt0(xa, sz00)), smndt0(xa))
% 0.21/0.74 = { by axiom 28 (mDistrib) R->L }
% 0.21/0.74 sdtpldt0(fresh14(true, true, xa, sz10, sz00), smndt0(xa))
% 0.21/0.74 = { by axiom 1 (mIntOne) R->L }
% 0.21/0.74 sdtpldt0(fresh14(aInteger0(sz10), true, xa, sz10, sz00), smndt0(xa))
% 0.21/0.74 = { by axiom 27 (mDistrib) R->L }
% 0.21/0.74 sdtpldt0(fresh20(aInteger0(sz00), true, xa, sz10, sz00), smndt0(xa))
% 0.21/0.74 = { by axiom 3 (mIntZero) }
% 0.21/0.74 sdtpldt0(fresh20(true, true, xa, sz10, sz00), smndt0(xa))
% 0.21/0.74 = { by axiom 25 (mDistrib) }
% 0.21/0.74 sdtpldt0(fresh21(aInteger0(xa), true, xa, sz10, sz00), smndt0(xa))
% 0.21/0.74 = { by axiom 2 (m__419) }
% 0.21/0.74 sdtpldt0(fresh21(true, true, xa, sz10, sz00), smndt0(xa))
% 0.21/0.74 = { by axiom 19 (mDistrib) }
% 0.21/0.74 sdtpldt0(sdtasdt0(xa, sdtpldt0(sz10, sz00)), smndt0(xa))
% 0.21/0.74 = { by axiom 17 (mAddZero) R->L }
% 0.21/0.74 sdtpldt0(sdtasdt0(xa, fresh3(aInteger0(sz10), true, sz10)), smndt0(xa))
% 0.21/0.74 = { by axiom 1 (mIntOne) }
% 0.21/0.74 sdtpldt0(sdtasdt0(xa, fresh3(true, true, sz10)), smndt0(xa))
% 0.21/0.74 = { by axiom 7 (mAddZero) }
% 0.21/0.74 sdtpldt0(sdtasdt0(xa, sz10), smndt0(xa))
% 0.21/0.74 = { by lemma 32 }
% 0.21/0.74 sdtpldt0(xa, smndt0(xa))
% 0.21/0.74 = { by lemma 30 }
% 0.21/0.74 sz00
% 0.21/0.74
% 0.21/0.74 Goal 1 (m__): tuple(sz00, sdtasdt0(xa, sz00)) = tuple(sdtasdt0(sz00, xa), sz00).
% 0.21/0.74 Proof:
% 0.21/0.74 tuple(sz00, sdtasdt0(xa, sz00))
% 0.21/0.74 = { by lemma 33 R->L }
% 0.21/0.74 tuple(sdtasdt0(sz00, xa), sdtasdt0(xa, sz00))
% 0.21/0.74 = { by lemma 31 R->L }
% 0.21/0.74 tuple(sdtasdt0(sz00, xa), sdtasdt0(sz00, xa))
% 0.21/0.74 = { by lemma 33 }
% 0.21/0.74 tuple(sdtasdt0(sz00, xa), sz00)
% 0.21/0.74 % SZS output end Proof
% 0.21/0.74
% 0.21/0.74 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------