/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 406 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 50 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 69 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 26 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 23 ms] (10) QTRS (11) QTRSRRRProof [EQUIVALENT, 13 ms] (12) QTRS (13) QTRSRRRProof [EQUIVALENT, 0 ms] (14) QTRS (15) RisEmptyProof [EQUIVALENT, 0 ms] (16) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(U11(tt, M, N)) -> mark(U12(tt, M, N)) active(U12(tt, M, N)) -> mark(s(plus(N, M))) active(U21(tt, M, N)) -> mark(U22(tt, M, N)) active(U22(tt, M, N)) -> mark(plus(x(N, M), N)) active(plus(N, 0)) -> mark(N) active(plus(N, s(M))) -> mark(U11(tt, M, N)) active(x(N, 0)) -> mark(0) active(x(N, s(M))) -> mark(U21(tt, M, N)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(U22(X1, X2, X3)) -> active(U22(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(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) 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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) U22(active(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, active(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, active(X3)) -> U22(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) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: active/1)YES( U11/3(YES,YES,YES) tt/0) mark/1)YES( U12/3(YES,YES,YES) s/1(YES) plus/2(YES,YES) U21/3(YES,YES,YES) U22/3(YES,YES,YES) x/2(YES,YES) 0/0) Quasi precedence: [U21_3, U22_3, x_2] > [U11_3, tt, U12_3, plus_2] > s_1 Status: U11_3: [3,2,1] tt: multiset status U12_3: [3,2,1] s_1: multiset status plus_2: [1,2] U21_3: [2,3,1] U22_3: [2,3,1] x_2: [2,1] 0: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(U12(tt, M, N)) -> mark(s(plus(N, M))) active(U22(tt, M, N)) -> mark(plus(x(N, M), N)) active(plus(N, 0)) -> mark(N) active(plus(N, s(M))) -> mark(U11(tt, M, N)) active(x(N, 0)) -> mark(0) active(x(N, s(M))) -> mark(U21(tt, M, N)) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(U11(tt, M, N)) -> mark(U12(tt, M, N)) active(U21(tt, M, N)) -> mark(U22(tt, M, N)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(U22(X1, X2, X3)) -> active(U22(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(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) 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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) U22(active(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, active(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, active(X3)) -> U22(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) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 POL(U12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U21(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 POL(U22(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 POL(active(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(plus(x_1, x_2)) = x_1 + 2*x_2 POL(s(x_1)) = x_1 POL(tt) = 0 POL(x(x_1, x_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: active(U11(tt, M, N)) -> mark(U12(tt, M, N)) active(U21(tt, M, N)) -> mark(U22(tt, M, N)) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(U22(X1, X2, X3)) -> active(U22(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(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) 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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) U22(active(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, active(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, active(X3)) -> U22(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) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 2 POL(U11(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U12(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 POL(U21(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3 POL(U22(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 POL(active(x_1)) = x_1 POL(mark(x_1)) = 2*x_1 POL(plus(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(s(x_1)) = 1 + 2*x_1 POL(tt) = 1 POL(x(x_1, x_2)) = x_1 + 2*x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(tt) -> active(tt) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(U22(X1, X2, X3)) -> active(U22(mark(X1), X2, X3)) mark(0) -> active(0) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(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) 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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) U22(active(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, active(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, active(X3)) -> U22(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) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 POL(U12(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U21(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U22(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(active(x_1)) = x_1 POL(mark(x_1)) = 2*x_1 POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2 POL(s(x_1)) = 2*x_1 POL(x(x_1, x_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, X3)) -> active(U11(mark(X1), X2, X3)) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(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) 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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) U22(active(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, active(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, active(X3)) -> U22(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) Q is empty. ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(active(x_1)) = 2 + x_1 POL(mark(x_1)) = 2*x_1 POL(plus(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = x_1 POL(x(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) s(active(X)) -> s(X) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(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) U22(active(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, active(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, active(X3)) -> U22(X1, X2, X3) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) ---------------------------------------- (10) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) s(mark(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) Q is empty. ---------------------------------------- (11) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U12(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U21(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(U22(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(active(x_1)) = x_1 POL(mark(x_1)) = 2*x_1 POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2 POL(s(x_1)) = x_1 POL(x(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) ---------------------------------------- (12) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) s(mark(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) Q is empty. ---------------------------------------- (13) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:mark_1 > x_2 > U22_3 > U21_3 > plus_2 > s_1 > U12_3 > U11_3 and weight map: mark_1=0 s_1=1 U11_3=0 U12_3=0 plus_2=0 U21_3=0 U22_3=0 x_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(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) s(mark(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(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) U22(mark(X1), X2, X3) -> U22(X1, X2, X3) U22(X1, mark(X2), X3) -> U22(X1, X2, X3) U22(X1, X2, mark(X3)) -> U22(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) ---------------------------------------- (14) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (15) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (16) YES