/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 400 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 88 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 25 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 20 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 0 ms] (10) QTRS (11) RisEmptyProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(U11(tt, N)) -> mark(N) active(U21(tt, M, N)) -> mark(s(plus(N, M))) active(U31(tt)) -> mark(0) active(U41(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) active(plus(N, 0)) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(x(N, 0)) -> mark(U31(isNat(N))) active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U31(X)) -> active(U31(mark(X))) mark(0) -> active(0) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(and(X1, X2)) -> active(and(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U21(mark(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, mark(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, mark(X3)) -> U21(X1, X2, X3) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: active/1)YES( U11/2(YES,YES) tt/0) mark/1)YES( U21/3(YES,YES,YES) s/1(YES) plus/2(YES,YES) U31/1(YES) 0/0) U41/3(YES,YES,YES) x/2(YES,YES) and/2(YES,YES) isNat/1)YES( Quasi precedence: [U41_3, x_2] > [U21_3, plus_2] > U11_2 [U41_3, x_2] > [U21_3, plus_2] > s_1 > and_2 [U41_3, x_2] > U31_1 > 0 > U11_2 [U41_3, x_2] > U31_1 > 0 > tt Status: U11_2: multiset status tt: multiset status U21_3: [3,2,1] s_1: multiset status plus_2: [1,2] U31_1: multiset status 0: multiset status U41_3: [2,3,1] x_2: [2,1] and_2: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(U11(tt, N)) -> mark(N) active(U21(tt, M, N)) -> mark(s(plus(N, M))) active(U31(tt)) -> mark(0) active(U41(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2))) active(plus(N, 0)) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(x(N, 0)) -> mark(U31(isNat(N))) active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N)) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U31(X)) -> active(U31(mark(X))) mark(0) -> active(0) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(and(X1, X2)) -> active(and(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U21(mark(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, mark(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, mark(X3)) -> U21(X1, X2, X3) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(U11(x_1, x_2)) = 1 + x_1 + x_2 POL(U21(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 POL(active(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 2 + x_1 POL(mark(x_1)) = 1 + 2*x_1 POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(s(x_1)) = x_1 POL(tt) = 2 POL(x(x_1, x_2)) = 2 + 2*x_1 + x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(0) -> active(0) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U21(mark(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, mark(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, mark(X3)) -> U21(X1, X2, X3) s(mark(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) U31(mark(X)) -> U31(X) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) isNat(mark(X)) -> isNat(X) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U31(X)) -> active(U31(mark(X))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(and(X1, X2)) -> active(and(mark(X1), X2)) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) s(active(X)) -> s(X) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U31(active(X)) -> U31(X) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNat(active(X)) -> isNat(X) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U21(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(active(x_1)) = x_1 POL(and(x_1, x_2)) = 1 + x_1 + x_2 POL(isNat(x_1)) = 2*x_1 POL(mark(x_1)) = 2 + 2*x_1 POL(plus(x_1, x_2)) = 2 + x_1 + x_2 POL(s(x_1)) = 2 + x_1 POL(x(x_1, x_2)) = 2 + x_1 + x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(s(X)) -> active(s(mark(X))) mark(and(X1, X2)) -> active(and(mark(X1), X2)) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U31(X)) -> active(U31(mark(X))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) s(active(X)) -> s(X) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U31(active(X)) -> U31(X) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNat(active(X)) -> isNat(X) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2)) = x_1 + x_2 POL(U21(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(U31(x_1)) = 1 + x_1 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(active(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 2*x_1 POL(mark(x_1)) = 1 + 2*x_1 POL(plus(x_1, x_2)) = 2 + x_1 + x_2 POL(s(x_1)) = 2*x_1 POL(x(x_1, x_2)) = 2 + x_1 + x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U31(X)) -> active(U31(mark(X))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) s(active(X)) -> s(X) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U31(active(X)) -> U31(X) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNat(active(X)) -> isNat(X) Q is empty. ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:active_1 > isNat_1 > and_2 > x_2 > U41_3 > U31_1 > plus_2 > s_1 > U21_3 > U11_2 and weight map: active_1=0 s_1=1 U31_1=1 isNat_1=1 U11_2=0 U21_3=0 plus_2=0 U41_3=0 x_2=0 and_2=0 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) s(active(X)) -> s(X) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U31(active(X)) -> U31(X) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNat(active(X)) -> isNat(X) ---------------------------------------- (10) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (11) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (12) YES