/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 68 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 17 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 18 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 0 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 2 ms] (10) QTRS (11) QTRSRRRProof [EQUIVALENT, 0 ms] (12) QTRS (13) QTRSRRRProof [EQUIVALENT, 14 ms] (14) QTRS (15) DependencyPairsProof [EQUIVALENT, 32 ms] (16) QDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) QDP (19) QReductionProof [EQUIVALENT, 0 ms] (20) QDP (21) MRRProof [EQUIVALENT, 0 ms] (22) QDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) QDP (25) QDPQMonotonicMRRProof [EQUIVALENT, 18 ms] (26) QDP (27) UsableRulesProof [EQUIVALENT, 0 ms] (28) QDP (29) QReductionProof [EQUIVALENT, 0 ms] (30) QDP (31) QDPSizeChangeProof [EQUIVALENT, 0 ms] (32) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__head(cons(X, XS)) -> mark(X) a__tail(cons(X, XS)) -> mark(XS) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds a__head(X) -> head(X) a__tail(X) -> tail(X) The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__head(x_1)) = 1 + x_1 POL(a__incr(x_1)) = 2*x_1 POL(a__nats) = 0 POL(a__odds) = 0 POL(a__pairs) = 0 POL(a__tail(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(head(x_1)) = 1 + x_1 POL(incr(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nats) = 0 POL(odds) = 0 POL(pairs) = 0 POL(s(x_1)) = x_1 POL(tail(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__head(cons(X, XS)) -> mark(X) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__tail(cons(X, XS)) -> mark(XS) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds a__head(X) -> head(X) a__tail(X) -> tail(X) The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__head(x_1)) = 1 + 2*x_1 POL(a__incr(x_1)) = 2*x_1 POL(a__nats) = 0 POL(a__odds) = 0 POL(a__pairs) = 0 POL(a__tail(x_1)) = 2*x_1 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(head(x_1)) = 1 + 2*x_1 POL(incr(x_1)) = 2*x_1 POL(mark(x_1)) = 2*x_1 POL(nats) = 0 POL(odds) = 0 POL(pairs) = 0 POL(s(x_1)) = x_1 POL(tail(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(head(X)) -> a__head(mark(X)) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__tail(cons(X, XS)) -> mark(XS) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds a__head(X) -> head(X) a__tail(X) -> tail(X) The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__head(x_1)) = 1 + 2*x_1 POL(a__incr(x_1)) = 2*x_1 POL(a__nats) = 0 POL(a__odds) = 0 POL(a__pairs) = 0 POL(a__tail(x_1)) = 2*x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(head(x_1)) = 2*x_1 POL(incr(x_1)) = 2*x_1 POL(mark(x_1)) = 2*x_1 POL(nats) = 0 POL(odds) = 0 POL(pairs) = 0 POL(s(x_1)) = x_1 POL(tail(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__head(X) -> head(X) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__tail(cons(X, XS)) -> mark(XS) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds a__tail(X) -> tail(X) The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__incr(x_1)) = 2*x_1 POL(a__nats) = 0 POL(a__odds) = 0 POL(a__pairs) = 0 POL(a__tail(x_1)) = 1 + 2*x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(incr(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nats) = 0 POL(odds) = 0 POL(pairs) = 0 POL(s(x_1)) = 2*x_1 POL(tail(x_1)) = 1 + 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__tail(cons(X, XS)) -> mark(XS) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds a__tail(X) -> tail(X) The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__incr(x_1)) = 2*x_1 POL(a__nats) = 0 POL(a__odds) = 0 POL(a__pairs) = 0 POL(a__tail(x_1)) = 1 + 2*x_1 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(incr(x_1)) = 2*x_1 POL(mark(x_1)) = 2*x_1 POL(nats) = 0 POL(odds) = 0 POL(pairs) = 0 POL(s(x_1)) = x_1 POL(tail(x_1)) = 1 + 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(tail(X)) -> a__tail(mark(X)) ---------------------------------------- (10) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds a__tail(X) -> tail(X) The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (11) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__incr(x_1)) = 2*x_1 POL(a__nats) = 0 POL(a__odds) = 0 POL(a__pairs) = 0 POL(a__tail(x_1)) = 1 + x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(incr(x_1)) = 2*x_1 POL(mark(x_1)) = 2*x_1 POL(nats) = 0 POL(odds) = 0 POL(pairs) = 0 POL(s(x_1)) = x_1 POL(tail(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__tail(X) -> tail(X) ---------------------------------------- (12) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (13) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__incr(x_1)) = 2*x_1 POL(a__nats) = 2 POL(a__odds) = 0 POL(a__pairs) = 0 POL(cons(x_1, x_2)) = x_1 + x_2 POL(incr(x_1)) = 2*x_1 POL(mark(x_1)) = 2*x_1 POL(nats) = 1 POL(odds) = 0 POL(pairs) = 0 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__nats -> nats ---------------------------------------- (14) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) ---------------------------------------- (15) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: A__ODDS -> A__INCR(a__pairs) A__ODDS -> A__PAIRS A__INCR(cons(X, XS)) -> MARK(X) MARK(nats) -> A__NATS MARK(incr(X)) -> A__INCR(mark(X)) MARK(incr(X)) -> MARK(X) MARK(pairs) -> A__PAIRS MARK(odds) -> A__ODDS MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: A__INCR(cons(X, XS)) -> MARK(X) MARK(incr(X)) -> A__INCR(mark(X)) MARK(incr(X)) -> MARK(X) MARK(odds) -> A__ODDS A__ODDS -> A__INCR(a__pairs) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) a__head(x0) a__tail(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. a__head(x0) a__tail(x0) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: A__INCR(cons(X, XS)) -> MARK(X) MARK(incr(X)) -> A__INCR(mark(X)) MARK(incr(X)) -> MARK(X) MARK(odds) -> A__ODDS A__ODDS -> A__INCR(a__pairs) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: MARK(odds) -> A__ODDS Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__INCR(x_1)) = x_1 POL(A__ODDS) = 2 POL(MARK(x_1)) = 2*x_1 POL(a__incr(x_1)) = x_1 POL(a__nats) = 0 POL(a__odds) = 2 POL(a__pairs) = 2 POL(cons(x_1, x_2)) = 2*x_1 + x_2 POL(incr(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nats) = 0 POL(odds) = 2 POL(pairs) = 2 POL(s(x_1)) = x_1 ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: A__INCR(cons(X, XS)) -> MARK(X) MARK(incr(X)) -> A__INCR(mark(X)) MARK(incr(X)) -> MARK(X) A__ODDS -> A__INCR(a__pairs) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(incr(X)) -> A__INCR(mark(X)) A__INCR(cons(X, XS)) -> MARK(X) MARK(incr(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(incr(X)) -> A__INCR(mark(X)) A__INCR(cons(X, XS)) -> MARK(X) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__INCR(x_1)) = 1 + 2*x_1 POL(MARK(x_1)) = 2*x_1 POL(a__incr(x_1)) = 2 + x_1 POL(a__nats) = 0 POL(a__odds) = 2 POL(a__pairs) = 0 POL(cons(x_1, x_2)) = x_1 POL(incr(x_1)) = 2 + x_1 POL(mark(x_1)) = x_1 POL(nats) = 0 POL(odds) = 2 POL(pairs) = 0 POL(s(x_1)) = 2 + x_1 ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> MARK(X1) The TRS R consists of the following rules: a__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> MARK(X1) R is empty. The set Q consists of the following terms: a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. a__nats a__pairs a__odds mark(nats) mark(incr(x0)) mark(pairs) mark(odds) mark(head(x0)) mark(tail(x0)) mark(cons(x0, x1)) mark(0) mark(s(x0)) a__incr(x0) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> MARK(X1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MARK(cons(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (32) YES