19.33/7.72 YES 19.53/7.75 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 19.53/7.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 19.53/7.75 19.53/7.75 19.53/7.75 Termination w.r.t. Q of the given QTRS could be proven: 19.53/7.75 19.53/7.75 (0) QTRS 19.53/7.75 (1) DependencyPairsProof [EQUIVALENT, 84 ms] 19.53/7.75 (2) QDP 19.53/7.75 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 19.53/7.75 (4) AND 19.53/7.75 (5) QDP 19.53/7.75 (6) UsableRulesProof [EQUIVALENT, 0 ms] 19.53/7.75 (7) QDP 19.53/7.75 (8) QReductionProof [EQUIVALENT, 0 ms] 19.53/7.75 (9) QDP 19.53/7.75 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 19.53/7.75 (11) YES 19.53/7.75 (12) QDP 19.53/7.75 (13) QDPOrderProof [EQUIVALENT, 1322 ms] 19.53/7.75 (14) QDP 19.53/7.75 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 19.53/7.75 (16) QDP 19.53/7.75 (17) UsableRulesProof [EQUIVALENT, 0 ms] 19.53/7.75 (18) QDP 19.53/7.75 (19) QReductionProof [EQUIVALENT, 0 ms] 19.53/7.75 (20) QDP 19.53/7.75 (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] 19.53/7.75 (22) YES 19.53/7.75 19.53/7.75 19.53/7.75 ---------------------------------------- 19.53/7.75 19.53/7.75 (0) 19.53/7.75 Obligation: 19.53/7.75 Q restricted rewrite system: 19.53/7.75 The TRS R consists of the following rules: 19.53/7.75 19.53/7.75 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.53/7.75 a__U12(tt) -> tt 19.53/7.75 a__U21(tt) -> tt 19.53/7.75 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.53/7.75 a__U32(tt) -> tt 19.53/7.75 a__U41(tt, N) -> mark(N) 19.53/7.75 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 19.53/7.75 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 19.53/7.75 a__U61(tt) -> 0 19.53/7.75 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 19.53/7.75 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 19.53/7.75 a__isNat(0) -> tt 19.53/7.75 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.53/7.75 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.53/7.75 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.53/7.75 a__plus(N, 0) -> a__U41(a__isNat(N), N) 19.53/7.75 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 19.53/7.75 a__x(N, 0) -> a__U61(a__isNat(N)) 19.53/7.75 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 19.53/7.75 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 19.53/7.75 mark(U12(X)) -> a__U12(mark(X)) 19.53/7.75 mark(isNat(X)) -> a__isNat(X) 19.53/7.75 mark(U21(X)) -> a__U21(mark(X)) 19.53/7.75 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 19.53/7.75 mark(U32(X)) -> a__U32(mark(X)) 19.53/7.75 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 19.53/7.75 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 19.53/7.75 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 19.53/7.75 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 19.53/7.75 mark(U61(X)) -> a__U61(mark(X)) 19.53/7.75 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 19.53/7.75 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 19.53/7.75 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 19.53/7.75 mark(tt) -> tt 19.53/7.75 mark(s(X)) -> s(mark(X)) 19.53/7.75 mark(0) -> 0 19.53/7.75 a__U11(X1, X2) -> U11(X1, X2) 19.53/7.75 a__U12(X) -> U12(X) 19.53/7.75 a__isNat(X) -> isNat(X) 19.53/7.75 a__U21(X) -> U21(X) 19.53/7.75 a__U31(X1, X2) -> U31(X1, X2) 19.53/7.75 a__U32(X) -> U32(X) 19.53/7.75 a__U41(X1, X2) -> U41(X1, X2) 19.53/7.75 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 19.53/7.75 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 19.53/7.75 a__plus(X1, X2) -> plus(X1, X2) 19.53/7.75 a__U61(X) -> U61(X) 19.53/7.75 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 19.53/7.75 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 19.53/7.75 a__x(X1, X2) -> x(X1, X2) 19.53/7.75 19.53/7.75 The set Q consists of the following terms: 19.53/7.75 19.53/7.75 mark(U11(x0, x1)) 19.53/7.75 mark(U12(x0)) 19.53/7.75 mark(isNat(x0)) 19.53/7.75 mark(U21(x0)) 19.53/7.75 mark(U31(x0, x1)) 19.53/7.75 mark(U32(x0)) 19.53/7.75 mark(U41(x0, x1)) 19.53/7.75 mark(U51(x0, x1, x2)) 19.53/7.75 mark(U52(x0, x1, x2)) 19.53/7.75 mark(plus(x0, x1)) 19.53/7.75 mark(U61(x0)) 19.53/7.75 mark(U71(x0, x1, x2)) 19.53/7.75 mark(U72(x0, x1, x2)) 19.53/7.75 mark(x(x0, x1)) 19.53/7.75 mark(tt) 19.53/7.75 mark(s(x0)) 19.53/7.75 mark(0) 19.53/7.75 a__U11(x0, x1) 19.53/7.75 a__U12(x0) 19.53/7.75 a__isNat(x0) 19.53/7.75 a__U21(x0) 19.53/7.75 a__U31(x0, x1) 19.53/7.75 a__U32(x0) 19.53/7.75 a__U41(x0, x1) 19.53/7.75 a__U51(x0, x1, x2) 19.53/7.75 a__U52(x0, x1, x2) 19.53/7.75 a__plus(x0, x1) 19.53/7.75 a__U61(x0) 19.53/7.75 a__U71(x0, x1, x2) 19.53/7.75 a__U72(x0, x1, x2) 19.53/7.75 a__x(x0, x1) 19.53/7.75 19.53/7.75 19.53/7.75 ---------------------------------------- 19.53/7.75 19.53/7.75 (1) DependencyPairsProof (EQUIVALENT) 19.53/7.75 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 19.53/7.75 ---------------------------------------- 19.53/7.75 19.53/7.75 (2) 19.53/7.75 Obligation: 19.53/7.75 Q DP problem: 19.53/7.75 The TRS P consists of the following rules: 19.53/7.75 19.53/7.75 A__U11(tt, V2) -> A__U12(a__isNat(V2)) 19.53/7.75 A__U11(tt, V2) -> A__ISNAT(V2) 19.53/7.75 A__U31(tt, V2) -> A__U32(a__isNat(V2)) 19.53/7.75 A__U31(tt, V2) -> A__ISNAT(V2) 19.53/7.75 A__U41(tt, N) -> MARK(N) 19.53/7.75 A__U51(tt, M, N) -> A__U52(a__isNat(N), M, N) 19.53/7.75 A__U51(tt, M, N) -> A__ISNAT(N) 19.53/7.75 A__U52(tt, M, N) -> A__PLUS(mark(N), mark(M)) 19.53/7.75 A__U52(tt, M, N) -> MARK(N) 19.53/7.75 A__U52(tt, M, N) -> MARK(M) 19.53/7.75 A__U71(tt, M, N) -> A__U72(a__isNat(N), M, N) 19.53/7.75 A__U71(tt, M, N) -> A__ISNAT(N) 19.53/7.75 A__U72(tt, M, N) -> A__PLUS(a__x(mark(N), mark(M)), mark(N)) 19.53/7.75 A__U72(tt, M, N) -> A__X(mark(N), mark(M)) 19.53/7.75 A__U72(tt, M, N) -> MARK(N) 19.53/7.75 A__U72(tt, M, N) -> MARK(M) 19.53/7.75 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 19.53/7.75 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 19.53/7.75 A__ISNAT(s(V1)) -> A__U21(a__isNat(V1)) 19.53/7.75 A__ISNAT(s(V1)) -> A__ISNAT(V1) 19.53/7.75 A__ISNAT(x(V1, V2)) -> A__U31(a__isNat(V1), V2) 19.53/7.75 A__ISNAT(x(V1, V2)) -> A__ISNAT(V1) 19.53/7.75 A__PLUS(N, 0) -> A__U41(a__isNat(N), N) 19.53/7.75 A__PLUS(N, 0) -> A__ISNAT(N) 19.53/7.75 A__PLUS(N, s(M)) -> A__U51(a__isNat(M), M, N) 19.53/7.75 A__PLUS(N, s(M)) -> A__ISNAT(M) 19.53/7.75 A__X(N, 0) -> A__U61(a__isNat(N)) 19.53/7.75 A__X(N, 0) -> A__ISNAT(N) 19.53/7.75 A__X(N, s(M)) -> A__U71(a__isNat(M), M, N) 19.53/7.75 A__X(N, s(M)) -> A__ISNAT(M) 19.53/7.75 MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) 19.53/7.75 MARK(U11(X1, X2)) -> MARK(X1) 19.53/7.75 MARK(U12(X)) -> A__U12(mark(X)) 19.53/7.75 MARK(U12(X)) -> MARK(X) 19.53/7.75 MARK(isNat(X)) -> A__ISNAT(X) 19.53/7.75 MARK(U21(X)) -> A__U21(mark(X)) 19.53/7.75 MARK(U21(X)) -> MARK(X) 19.53/7.75 MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) 19.53/7.75 MARK(U31(X1, X2)) -> MARK(X1) 19.53/7.75 MARK(U32(X)) -> A__U32(mark(X)) 19.53/7.75 MARK(U32(X)) -> MARK(X) 19.53/7.75 MARK(U41(X1, X2)) -> A__U41(mark(X1), X2) 19.53/7.75 MARK(U41(X1, X2)) -> MARK(X1) 19.53/7.75 MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) 19.53/7.75 MARK(U51(X1, X2, X3)) -> MARK(X1) 19.53/7.75 MARK(U52(X1, X2, X3)) -> A__U52(mark(X1), X2, X3) 19.53/7.75 MARK(U52(X1, X2, X3)) -> MARK(X1) 19.53/7.75 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 19.53/7.75 MARK(plus(X1, X2)) -> MARK(X1) 19.53/7.75 MARK(plus(X1, X2)) -> MARK(X2) 19.53/7.75 MARK(U61(X)) -> A__U61(mark(X)) 19.53/7.75 MARK(U61(X)) -> MARK(X) 19.53/7.75 MARK(U71(X1, X2, X3)) -> A__U71(mark(X1), X2, X3) 19.53/7.75 MARK(U71(X1, X2, X3)) -> MARK(X1) 19.53/7.75 MARK(U72(X1, X2, X3)) -> A__U72(mark(X1), X2, X3) 19.53/7.75 MARK(U72(X1, X2, X3)) -> MARK(X1) 19.53/7.75 MARK(x(X1, X2)) -> A__X(mark(X1), mark(X2)) 19.53/7.75 MARK(x(X1, X2)) -> MARK(X1) 19.53/7.75 MARK(x(X1, X2)) -> MARK(X2) 19.53/7.75 MARK(s(X)) -> MARK(X) 19.53/7.75 19.53/7.75 The TRS R consists of the following rules: 19.53/7.75 19.53/7.75 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.53/7.75 a__U12(tt) -> tt 19.53/7.75 a__U21(tt) -> tt 19.53/7.75 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.53/7.75 a__U32(tt) -> tt 19.53/7.75 a__U41(tt, N) -> mark(N) 19.53/7.75 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 19.53/7.75 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 19.53/7.75 a__U61(tt) -> 0 19.53/7.75 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 19.53/7.75 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 19.53/7.75 a__isNat(0) -> tt 19.53/7.75 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.53/7.75 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.53/7.75 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.53/7.75 a__plus(N, 0) -> a__U41(a__isNat(N), N) 19.53/7.75 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 19.53/7.75 a__x(N, 0) -> a__U61(a__isNat(N)) 19.53/7.75 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 19.53/7.75 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 19.53/7.75 mark(U12(X)) -> a__U12(mark(X)) 19.53/7.75 mark(isNat(X)) -> a__isNat(X) 19.53/7.75 mark(U21(X)) -> a__U21(mark(X)) 19.53/7.75 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 19.53/7.75 mark(U32(X)) -> a__U32(mark(X)) 19.53/7.75 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 19.53/7.75 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 19.53/7.75 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 19.53/7.75 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 19.53/7.75 mark(U61(X)) -> a__U61(mark(X)) 19.53/7.75 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 19.53/7.75 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 19.53/7.75 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 19.53/7.75 mark(tt) -> tt 19.53/7.75 mark(s(X)) -> s(mark(X)) 19.53/7.75 mark(0) -> 0 19.53/7.75 a__U11(X1, X2) -> U11(X1, X2) 19.53/7.75 a__U12(X) -> U12(X) 19.53/7.75 a__isNat(X) -> isNat(X) 19.53/7.75 a__U21(X) -> U21(X) 19.53/7.75 a__U31(X1, X2) -> U31(X1, X2) 19.53/7.75 a__U32(X) -> U32(X) 19.53/7.75 a__U41(X1, X2) -> U41(X1, X2) 19.53/7.75 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 19.53/7.75 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 19.53/7.75 a__plus(X1, X2) -> plus(X1, X2) 19.53/7.75 a__U61(X) -> U61(X) 19.53/7.75 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 19.53/7.75 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 19.53/7.75 a__x(X1, X2) -> x(X1, X2) 19.53/7.75 19.53/7.75 The set Q consists of the following terms: 19.53/7.75 19.53/7.75 mark(U11(x0, x1)) 19.53/7.75 mark(U12(x0)) 19.53/7.75 mark(isNat(x0)) 19.53/7.75 mark(U21(x0)) 19.53/7.75 mark(U31(x0, x1)) 19.53/7.75 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (3) DependencyGraphProof (EQUIVALENT) 19.59/7.77 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 17 less nodes. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (4) 19.59/7.77 Complex Obligation (AND) 19.59/7.77 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (5) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 A__U11(tt, V2) -> A__ISNAT(V2) 19.59/7.77 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 19.59/7.77 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 A__ISNAT(s(V1)) -> A__ISNAT(V1) 19.59/7.77 A__ISNAT(x(V1, V2)) -> A__U31(a__isNat(V1), V2) 19.59/7.77 A__U31(tt, V2) -> A__ISNAT(V2) 19.59/7.77 A__ISNAT(x(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 19.59/7.77 The TRS R consists of the following rules: 19.59/7.77 19.59/7.77 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.59/7.77 a__U12(tt) -> tt 19.59/7.77 a__U21(tt) -> tt 19.59/7.77 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.59/7.77 a__U32(tt) -> tt 19.59/7.77 a__U41(tt, N) -> mark(N) 19.59/7.77 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 19.59/7.77 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 19.59/7.77 a__U61(tt) -> 0 19.59/7.77 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 19.59/7.77 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 19.59/7.77 a__isNat(0) -> tt 19.59/7.77 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.59/7.77 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.59/7.77 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.59/7.77 a__plus(N, 0) -> a__U41(a__isNat(N), N) 19.59/7.77 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 19.59/7.77 a__x(N, 0) -> a__U61(a__isNat(N)) 19.59/7.77 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 19.59/7.77 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 19.59/7.77 mark(U12(X)) -> a__U12(mark(X)) 19.59/7.77 mark(isNat(X)) -> a__isNat(X) 19.59/7.77 mark(U21(X)) -> a__U21(mark(X)) 19.59/7.77 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 19.59/7.77 mark(U32(X)) -> a__U32(mark(X)) 19.59/7.77 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 19.59/7.77 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 19.59/7.77 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 19.59/7.77 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 19.59/7.77 mark(U61(X)) -> a__U61(mark(X)) 19.59/7.77 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 19.59/7.77 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 19.59/7.77 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 19.59/7.77 mark(tt) -> tt 19.59/7.77 mark(s(X)) -> s(mark(X)) 19.59/7.77 mark(0) -> 0 19.59/7.77 a__U11(X1, X2) -> U11(X1, X2) 19.59/7.77 a__U12(X) -> U12(X) 19.59/7.77 a__isNat(X) -> isNat(X) 19.59/7.77 a__U21(X) -> U21(X) 19.59/7.77 a__U31(X1, X2) -> U31(X1, X2) 19.59/7.77 a__U32(X) -> U32(X) 19.59/7.77 a__U41(X1, X2) -> U41(X1, X2) 19.59/7.77 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 19.59/7.77 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 19.59/7.77 a__plus(X1, X2) -> plus(X1, X2) 19.59/7.77 a__U61(X) -> U61(X) 19.59/7.77 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 19.59/7.77 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 19.59/7.77 a__x(X1, X2) -> x(X1, X2) 19.59/7.77 19.59/7.77 The set Q consists of the following terms: 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (6) UsableRulesProof (EQUIVALENT) 19.59/7.77 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. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (7) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 A__U11(tt, V2) -> A__ISNAT(V2) 19.59/7.77 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 19.59/7.77 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 A__ISNAT(s(V1)) -> A__ISNAT(V1) 19.59/7.77 A__ISNAT(x(V1, V2)) -> A__U31(a__isNat(V1), V2) 19.59/7.77 A__U31(tt, V2) -> A__ISNAT(V2) 19.59/7.77 A__ISNAT(x(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 19.59/7.77 The TRS R consists of the following rules: 19.59/7.77 19.59/7.77 a__isNat(0) -> tt 19.59/7.77 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.59/7.77 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.59/7.77 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.59/7.77 a__isNat(X) -> isNat(X) 19.59/7.77 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.59/7.77 a__U31(X1, X2) -> U31(X1, X2) 19.59/7.77 a__U32(tt) -> tt 19.59/7.77 a__U32(X) -> U32(X) 19.59/7.77 a__U21(tt) -> tt 19.59/7.77 a__U21(X) -> U21(X) 19.59/7.77 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.59/7.77 a__U11(X1, X2) -> U11(X1, X2) 19.59/7.77 a__U12(tt) -> tt 19.59/7.77 a__U12(X) -> U12(X) 19.59/7.77 19.59/7.77 The set Q consists of the following terms: 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (8) QReductionProof (EQUIVALENT) 19.59/7.77 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (9) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 A__U11(tt, V2) -> A__ISNAT(V2) 19.59/7.77 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 19.59/7.77 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 A__ISNAT(s(V1)) -> A__ISNAT(V1) 19.59/7.77 A__ISNAT(x(V1, V2)) -> A__U31(a__isNat(V1), V2) 19.59/7.77 A__U31(tt, V2) -> A__ISNAT(V2) 19.59/7.77 A__ISNAT(x(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 19.59/7.77 The TRS R consists of the following rules: 19.59/7.77 19.59/7.77 a__isNat(0) -> tt 19.59/7.77 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.59/7.77 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.59/7.77 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.59/7.77 a__isNat(X) -> isNat(X) 19.59/7.77 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.59/7.77 a__U31(X1, X2) -> U31(X1, X2) 19.59/7.77 a__U32(tt) -> tt 19.59/7.77 a__U32(X) -> U32(X) 19.59/7.77 a__U21(tt) -> tt 19.59/7.77 a__U21(X) -> U21(X) 19.59/7.77 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.59/7.77 a__U11(X1, X2) -> U11(X1, X2) 19.59/7.77 a__U12(tt) -> tt 19.59/7.77 a__U12(X) -> U12(X) 19.59/7.77 19.59/7.77 The set Q consists of the following terms: 19.59/7.77 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (10) QDPSizeChangeProof (EQUIVALENT) 19.59/7.77 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. 19.59/7.77 19.59/7.77 From the DPs we obtained the following set of size-change graphs: 19.59/7.77 *A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 19.59/7.77 The graph contains the following edges 1 > 2 19.59/7.77 19.59/7.77 19.59/7.77 *A__ISNAT(x(V1, V2)) -> A__U31(a__isNat(V1), V2) 19.59/7.77 The graph contains the following edges 1 > 2 19.59/7.77 19.59/7.77 19.59/7.77 *A__U11(tt, V2) -> A__ISNAT(V2) 19.59/7.77 The graph contains the following edges 2 >= 1 19.59/7.77 19.59/7.77 19.59/7.77 *A__U31(tt, V2) -> A__ISNAT(V2) 19.59/7.77 The graph contains the following edges 2 >= 1 19.59/7.77 19.59/7.77 19.59/7.77 *A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 *A__ISNAT(s(V1)) -> A__ISNAT(V1) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 *A__ISNAT(x(V1, V2)) -> A__ISNAT(V1) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (11) 19.59/7.77 YES 19.59/7.77 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (12) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 MARK(U11(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U12(X)) -> MARK(X) 19.59/7.77 MARK(U21(X)) -> MARK(X) 19.59/7.77 MARK(U31(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U32(X)) -> MARK(X) 19.59/7.77 MARK(U41(X1, X2)) -> A__U41(mark(X1), X2) 19.59/7.77 A__U41(tt, N) -> MARK(N) 19.59/7.77 MARK(U41(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) 19.59/7.77 A__U51(tt, M, N) -> A__U52(a__isNat(N), M, N) 19.59/7.77 A__U52(tt, M, N) -> A__PLUS(mark(N), mark(M)) 19.59/7.77 A__PLUS(N, 0) -> A__U41(a__isNat(N), N) 19.59/7.77 A__PLUS(N, s(M)) -> A__U51(a__isNat(M), M, N) 19.59/7.77 A__U52(tt, M, N) -> MARK(N) 19.59/7.77 MARK(U51(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(U52(X1, X2, X3)) -> A__U52(mark(X1), X2, X3) 19.59/7.77 A__U52(tt, M, N) -> MARK(M) 19.59/7.77 MARK(U52(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 19.59/7.77 MARK(plus(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(plus(X1, X2)) -> MARK(X2) 19.59/7.77 MARK(U61(X)) -> MARK(X) 19.59/7.77 MARK(U71(X1, X2, X3)) -> A__U71(mark(X1), X2, X3) 19.59/7.77 A__U71(tt, M, N) -> A__U72(a__isNat(N), M, N) 19.59/7.77 A__U72(tt, M, N) -> A__PLUS(a__x(mark(N), mark(M)), mark(N)) 19.59/7.77 A__U72(tt, M, N) -> A__X(mark(N), mark(M)) 19.59/7.77 A__X(N, s(M)) -> A__U71(a__isNat(M), M, N) 19.59/7.77 A__U72(tt, M, N) -> MARK(N) 19.59/7.77 MARK(U71(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(U72(X1, X2, X3)) -> A__U72(mark(X1), X2, X3) 19.59/7.77 A__U72(tt, M, N) -> MARK(M) 19.59/7.77 MARK(U72(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(x(X1, X2)) -> A__X(mark(X1), mark(X2)) 19.59/7.77 MARK(x(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(x(X1, X2)) -> MARK(X2) 19.59/7.77 MARK(s(X)) -> MARK(X) 19.59/7.77 19.59/7.77 The TRS R consists of the following rules: 19.59/7.77 19.59/7.77 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.59/7.77 a__U12(tt) -> tt 19.59/7.77 a__U21(tt) -> tt 19.59/7.77 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.59/7.77 a__U32(tt) -> tt 19.59/7.77 a__U41(tt, N) -> mark(N) 19.59/7.77 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 19.59/7.77 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 19.59/7.77 a__U61(tt) -> 0 19.59/7.77 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 19.59/7.77 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 19.59/7.77 a__isNat(0) -> tt 19.59/7.77 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.59/7.77 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.59/7.77 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.59/7.77 a__plus(N, 0) -> a__U41(a__isNat(N), N) 19.59/7.77 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 19.59/7.77 a__x(N, 0) -> a__U61(a__isNat(N)) 19.59/7.77 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 19.59/7.77 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 19.59/7.77 mark(U12(X)) -> a__U12(mark(X)) 19.59/7.77 mark(isNat(X)) -> a__isNat(X) 19.59/7.77 mark(U21(X)) -> a__U21(mark(X)) 19.59/7.77 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 19.59/7.77 mark(U32(X)) -> a__U32(mark(X)) 19.59/7.77 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 19.59/7.77 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 19.59/7.77 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 19.59/7.77 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 19.59/7.77 mark(U61(X)) -> a__U61(mark(X)) 19.59/7.77 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 19.59/7.77 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 19.59/7.77 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 19.59/7.77 mark(tt) -> tt 19.59/7.77 mark(s(X)) -> s(mark(X)) 19.59/7.77 mark(0) -> 0 19.59/7.77 a__U11(X1, X2) -> U11(X1, X2) 19.59/7.77 a__U12(X) -> U12(X) 19.59/7.77 a__isNat(X) -> isNat(X) 19.59/7.77 a__U21(X) -> U21(X) 19.59/7.77 a__U31(X1, X2) -> U31(X1, X2) 19.59/7.77 a__U32(X) -> U32(X) 19.59/7.77 a__U41(X1, X2) -> U41(X1, X2) 19.59/7.77 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 19.59/7.77 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 19.59/7.77 a__plus(X1, X2) -> plus(X1, X2) 19.59/7.77 a__U61(X) -> U61(X) 19.59/7.77 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 19.59/7.77 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 19.59/7.77 a__x(X1, X2) -> x(X1, X2) 19.59/7.77 19.59/7.77 The set Q consists of the following terms: 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (13) QDPOrderProof (EQUIVALENT) 19.59/7.77 We use the reduction pair processor [LPAR04,JAR06]. 19.59/7.77 19.59/7.77 19.59/7.77 The following pairs can be oriented strictly and are deleted. 19.59/7.77 19.59/7.77 MARK(U41(X1, X2)) -> A__U41(mark(X1), X2) 19.59/7.77 MARK(U41(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) 19.59/7.77 A__U52(tt, M, N) -> A__PLUS(mark(N), mark(M)) 19.59/7.77 A__PLUS(N, 0) -> A__U41(a__isNat(N), N) 19.59/7.77 A__PLUS(N, s(M)) -> A__U51(a__isNat(M), M, N) 19.59/7.77 A__U52(tt, M, N) -> MARK(N) 19.59/7.77 MARK(U51(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(U52(X1, X2, X3)) -> A__U52(mark(X1), X2, X3) 19.59/7.77 A__U52(tt, M, N) -> MARK(M) 19.59/7.77 MARK(U52(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 19.59/7.77 MARK(plus(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(plus(X1, X2)) -> MARK(X2) 19.59/7.77 MARK(U71(X1, X2, X3)) -> A__U71(mark(X1), X2, X3) 19.59/7.77 A__U72(tt, M, N) -> A__PLUS(a__x(mark(N), mark(M)), mark(N)) 19.59/7.77 A__U72(tt, M, N) -> A__X(mark(N), mark(M)) 19.59/7.77 A__X(N, s(M)) -> A__U71(a__isNat(M), M, N) 19.59/7.77 A__U72(tt, M, N) -> MARK(N) 19.59/7.77 MARK(U71(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(U72(X1, X2, X3)) -> A__U72(mark(X1), X2, X3) 19.59/7.77 A__U72(tt, M, N) -> MARK(M) 19.59/7.77 MARK(U72(X1, X2, X3)) -> MARK(X1) 19.59/7.77 MARK(x(X1, X2)) -> A__X(mark(X1), mark(X2)) 19.59/7.77 MARK(x(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(x(X1, X2)) -> MARK(X2) 19.59/7.77 MARK(s(X)) -> MARK(X) 19.59/7.77 The remaining pairs can at least be oriented weakly. 19.59/7.77 Used ordering: Combined order from the following AFS and order. 19.59/7.77 MARK(x1) = MARK(x1) 19.59/7.77 19.59/7.77 U11(x1, x2) = x1 19.59/7.77 19.59/7.77 U12(x1) = x1 19.59/7.77 19.59/7.77 U21(x1) = x1 19.59/7.77 19.59/7.77 U31(x1, x2) = x1 19.59/7.77 19.59/7.77 U32(x1) = x1 19.59/7.77 19.59/7.77 U41(x1, x2) = U41(x1, x2) 19.59/7.77 19.59/7.77 A__U41(x1, x2) = A__U41(x2) 19.59/7.77 19.59/7.77 mark(x1) = x1 19.59/7.77 19.59/7.77 tt = tt 19.59/7.77 19.59/7.77 U51(x1, x2, x3) = U51(x1, x2, x3) 19.59/7.77 19.59/7.77 A__U51(x1, x2, x3) = A__U51(x1, x2, x3) 19.59/7.77 19.59/7.77 A__U52(x1, x2, x3) = A__U52(x1, x2, x3) 19.59/7.77 19.59/7.77 a__isNat(x1) = a__isNat 19.59/7.77 19.59/7.77 A__PLUS(x1, x2) = A__PLUS(x1, x2) 19.59/7.77 19.59/7.77 0 = 0 19.59/7.77 19.59/7.77 s(x1) = s(x1) 19.59/7.77 19.59/7.77 U52(x1, x2, x3) = U52(x1, x2, x3) 19.59/7.77 19.59/7.77 plus(x1, x2) = plus(x1, x2) 19.59/7.77 19.59/7.77 U61(x1) = x1 19.59/7.77 19.59/7.77 U71(x1, x2, x3) = U71(x1, x2, x3) 19.59/7.77 19.59/7.77 A__U71(x1, x2, x3) = A__U71(x1, x2, x3) 19.59/7.77 19.59/7.77 A__U72(x1, x2, x3) = A__U72(x1, x2, x3) 19.59/7.77 19.59/7.77 a__x(x1, x2) = a__x(x1, x2) 19.59/7.77 19.59/7.77 A__X(x1, x2) = A__X(x1, x2) 19.59/7.77 19.59/7.77 U72(x1, x2, x3) = U72(x1, x2, x3) 19.59/7.77 19.59/7.77 x(x1, x2) = x(x1, x2) 19.59/7.77 19.59/7.77 a__U11(x1, x2) = x1 19.59/7.77 19.59/7.77 a__U12(x1) = x1 19.59/7.77 19.59/7.77 isNat(x1) = isNat 19.59/7.77 19.59/7.77 a__U21(x1) = x1 19.59/7.77 19.59/7.77 a__U31(x1, x2) = x1 19.59/7.77 19.59/7.77 a__U32(x1) = x1 19.59/7.77 19.59/7.77 a__U41(x1, x2) = a__U41(x1, x2) 19.59/7.77 19.59/7.77 a__plus(x1, x2) = a__plus(x1, x2) 19.59/7.77 19.59/7.77 a__U71(x1, x2, x3) = a__U71(x1, x2, x3) 19.59/7.77 19.59/7.77 a__U72(x1, x2, x3) = a__U72(x1, x2, x3) 19.59/7.77 19.59/7.77 a__U51(x1, x2, x3) = a__U51(x1, x2, x3) 19.59/7.77 19.59/7.77 a__U52(x1, x2, x3) = a__U52(x1, x2, x3) 19.59/7.77 19.59/7.77 a__U61(x1) = x1 19.59/7.77 19.59/7.77 19.59/7.77 Recursive path order with status [RPO]. 19.59/7.77 Quasi-Precedence: [U71_3, A__U71_3, A__U72_3, a__x_2, A__X_2, U72_3, x_2, a__U71_3, a__U72_3] > [U51_3, U52_3, plus_2, a__plus_2, a__U51_3, a__U52_3] > [tt, a__isNat, s_1, isNat] > [MARK_1, A__U41_1, A__U51_3, A__U52_3, A__PLUS_2, 0] > [U41_2, a__U41_2] 19.59/7.77 19.59/7.77 Status: MARK_1: multiset status 19.59/7.77 U41_2: multiset status 19.59/7.77 A__U41_1: multiset status 19.59/7.77 tt: multiset status 19.59/7.77 U51_3: multiset status 19.59/7.77 A__U51_3: multiset status 19.59/7.77 A__U52_3: multiset status 19.59/7.77 a__isNat: multiset status 19.59/7.77 A__PLUS_2: multiset status 19.59/7.77 0: multiset status 19.59/7.77 s_1: multiset status 19.59/7.77 U52_3: multiset status 19.59/7.77 plus_2: multiset status 19.59/7.77 U71_3: [2,3,1] 19.59/7.77 A__U71_3: [2,3,1] 19.59/7.77 A__U72_3: [2,3,1] 19.59/7.77 a__x_2: [2,1] 19.59/7.77 A__X_2: [2,1] 19.59/7.77 U72_3: [2,3,1] 19.59/7.77 x_2: [2,1] 19.59/7.77 isNat: multiset status 19.59/7.77 a__U41_2: multiset status 19.59/7.77 a__plus_2: multiset status 19.59/7.77 a__U71_3: [2,3,1] 19.59/7.77 a__U72_3: [2,3,1] 19.59/7.77 a__U51_3: multiset status 19.59/7.77 a__U52_3: multiset status 19.59/7.77 19.59/7.77 19.59/7.77 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 19.59/7.77 19.59/7.77 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 19.59/7.77 mark(U12(X)) -> a__U12(mark(X)) 19.59/7.77 mark(isNat(X)) -> a__isNat(X) 19.59/7.77 mark(U21(X)) -> a__U21(mark(X)) 19.59/7.77 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 19.59/7.77 mark(U32(X)) -> a__U32(mark(X)) 19.59/7.77 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 19.59/7.77 a__U41(tt, N) -> mark(N) 19.59/7.77 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 19.59/7.77 a__plus(N, 0) -> a__U41(a__isNat(N), N) 19.59/7.77 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 19.59/7.77 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 19.59/7.77 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 19.59/7.77 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 19.59/7.77 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 19.59/7.77 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 19.59/7.77 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 19.59/7.77 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 19.59/7.77 mark(U61(X)) -> a__U61(mark(X)) 19.59/7.77 mark(tt) -> tt 19.59/7.77 mark(s(X)) -> s(mark(X)) 19.59/7.77 mark(0) -> 0 19.59/7.77 a__isNat(0) -> tt 19.59/7.77 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.59/7.77 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.59/7.77 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.59/7.77 a__isNat(X) -> isNat(X) 19.59/7.77 a__x(N, 0) -> a__U61(a__isNat(N)) 19.59/7.77 a__x(X1, X2) -> x(X1, X2) 19.59/7.77 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 19.59/7.77 a__plus(X1, X2) -> plus(X1, X2) 19.59/7.77 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.59/7.77 a__U11(X1, X2) -> U11(X1, X2) 19.59/7.77 a__U12(tt) -> tt 19.59/7.77 a__U12(X) -> U12(X) 19.59/7.77 a__U21(tt) -> tt 19.59/7.77 a__U21(X) -> U21(X) 19.59/7.77 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.59/7.77 a__U31(X1, X2) -> U31(X1, X2) 19.59/7.77 a__U32(tt) -> tt 19.59/7.77 a__U32(X) -> U32(X) 19.59/7.77 a__U41(X1, X2) -> U41(X1, X2) 19.59/7.77 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 19.59/7.77 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 19.59/7.77 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 19.59/7.77 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 19.59/7.77 a__U61(tt) -> 0 19.59/7.77 a__U61(X) -> U61(X) 19.59/7.77 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 19.59/7.77 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 19.59/7.77 19.59/7.77 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (14) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 MARK(U11(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U12(X)) -> MARK(X) 19.59/7.77 MARK(U21(X)) -> MARK(X) 19.59/7.77 MARK(U31(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U32(X)) -> MARK(X) 19.59/7.77 A__U41(tt, N) -> MARK(N) 19.59/7.77 A__U51(tt, M, N) -> A__U52(a__isNat(N), M, N) 19.59/7.77 MARK(U61(X)) -> MARK(X) 19.59/7.77 A__U71(tt, M, N) -> A__U72(a__isNat(N), M, N) 19.59/7.77 19.59/7.77 The TRS R consists of the following rules: 19.59/7.77 19.59/7.77 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.59/7.77 a__U12(tt) -> tt 19.59/7.77 a__U21(tt) -> tt 19.59/7.77 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.59/7.77 a__U32(tt) -> tt 19.59/7.77 a__U41(tt, N) -> mark(N) 19.59/7.77 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 19.59/7.77 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 19.59/7.77 a__U61(tt) -> 0 19.59/7.77 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 19.59/7.77 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 19.59/7.77 a__isNat(0) -> tt 19.59/7.77 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.59/7.77 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.59/7.77 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.59/7.77 a__plus(N, 0) -> a__U41(a__isNat(N), N) 19.59/7.77 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 19.59/7.77 a__x(N, 0) -> a__U61(a__isNat(N)) 19.59/7.77 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 19.59/7.77 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 19.59/7.77 mark(U12(X)) -> a__U12(mark(X)) 19.59/7.77 mark(isNat(X)) -> a__isNat(X) 19.59/7.77 mark(U21(X)) -> a__U21(mark(X)) 19.59/7.77 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 19.59/7.77 mark(U32(X)) -> a__U32(mark(X)) 19.59/7.77 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 19.59/7.77 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 19.59/7.77 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 19.59/7.77 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 19.59/7.77 mark(U61(X)) -> a__U61(mark(X)) 19.59/7.77 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 19.59/7.77 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 19.59/7.77 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 19.59/7.77 mark(tt) -> tt 19.59/7.77 mark(s(X)) -> s(mark(X)) 19.59/7.77 mark(0) -> 0 19.59/7.77 a__U11(X1, X2) -> U11(X1, X2) 19.59/7.77 a__U12(X) -> U12(X) 19.59/7.77 a__isNat(X) -> isNat(X) 19.59/7.77 a__U21(X) -> U21(X) 19.59/7.77 a__U31(X1, X2) -> U31(X1, X2) 19.59/7.77 a__U32(X) -> U32(X) 19.59/7.77 a__U41(X1, X2) -> U41(X1, X2) 19.59/7.77 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 19.59/7.77 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 19.59/7.77 a__plus(X1, X2) -> plus(X1, X2) 19.59/7.77 a__U61(X) -> U61(X) 19.59/7.77 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 19.59/7.77 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 19.59/7.77 a__x(X1, X2) -> x(X1, X2) 19.59/7.77 19.59/7.77 The set Q consists of the following terms: 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (15) DependencyGraphProof (EQUIVALENT) 19.59/7.77 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (16) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 MARK(U12(X)) -> MARK(X) 19.59/7.77 MARK(U11(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U21(X)) -> MARK(X) 19.59/7.77 MARK(U31(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U32(X)) -> MARK(X) 19.59/7.77 MARK(U61(X)) -> MARK(X) 19.59/7.77 19.59/7.77 The TRS R consists of the following rules: 19.59/7.77 19.59/7.77 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 19.59/7.77 a__U12(tt) -> tt 19.59/7.77 a__U21(tt) -> tt 19.59/7.77 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 19.59/7.77 a__U32(tt) -> tt 19.59/7.77 a__U41(tt, N) -> mark(N) 19.59/7.77 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 19.59/7.77 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 19.59/7.77 a__U61(tt) -> 0 19.59/7.77 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 19.59/7.77 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 19.59/7.77 a__isNat(0) -> tt 19.59/7.77 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 19.59/7.77 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 19.59/7.77 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 19.59/7.77 a__plus(N, 0) -> a__U41(a__isNat(N), N) 19.59/7.77 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 19.59/7.77 a__x(N, 0) -> a__U61(a__isNat(N)) 19.59/7.77 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 19.59/7.77 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 19.59/7.77 mark(U12(X)) -> a__U12(mark(X)) 19.59/7.77 mark(isNat(X)) -> a__isNat(X) 19.59/7.77 mark(U21(X)) -> a__U21(mark(X)) 19.59/7.77 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 19.59/7.77 mark(U32(X)) -> a__U32(mark(X)) 19.59/7.77 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 19.59/7.77 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 19.59/7.77 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 19.59/7.77 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 19.59/7.77 mark(U61(X)) -> a__U61(mark(X)) 19.59/7.77 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 19.59/7.77 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 19.59/7.77 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 19.59/7.77 mark(tt) -> tt 19.59/7.77 mark(s(X)) -> s(mark(X)) 19.59/7.77 mark(0) -> 0 19.59/7.77 a__U11(X1, X2) -> U11(X1, X2) 19.59/7.77 a__U12(X) -> U12(X) 19.59/7.77 a__isNat(X) -> isNat(X) 19.59/7.77 a__U21(X) -> U21(X) 19.59/7.77 a__U31(X1, X2) -> U31(X1, X2) 19.59/7.77 a__U32(X) -> U32(X) 19.59/7.77 a__U41(X1, X2) -> U41(X1, X2) 19.59/7.77 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 19.59/7.77 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 19.59/7.77 a__plus(X1, X2) -> plus(X1, X2) 19.59/7.77 a__U61(X) -> U61(X) 19.59/7.77 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 19.59/7.77 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 19.59/7.77 a__x(X1, X2) -> x(X1, X2) 19.59/7.77 19.59/7.77 The set Q consists of the following terms: 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (17) UsableRulesProof (EQUIVALENT) 19.59/7.77 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. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (18) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 MARK(U12(X)) -> MARK(X) 19.59/7.77 MARK(U11(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U21(X)) -> MARK(X) 19.59/7.77 MARK(U31(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U32(X)) -> MARK(X) 19.59/7.77 MARK(U61(X)) -> MARK(X) 19.59/7.77 19.59/7.77 R is empty. 19.59/7.77 The set Q consists of the following terms: 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (19) QReductionProof (EQUIVALENT) 19.59/7.77 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 19.59/7.77 19.59/7.77 mark(U11(x0, x1)) 19.59/7.77 mark(U12(x0)) 19.59/7.77 mark(isNat(x0)) 19.59/7.77 mark(U21(x0)) 19.59/7.77 mark(U31(x0, x1)) 19.59/7.77 mark(U32(x0)) 19.59/7.77 mark(U41(x0, x1)) 19.59/7.77 mark(U51(x0, x1, x2)) 19.59/7.77 mark(U52(x0, x1, x2)) 19.59/7.77 mark(plus(x0, x1)) 19.59/7.77 mark(U61(x0)) 19.59/7.77 mark(U71(x0, x1, x2)) 19.59/7.77 mark(U72(x0, x1, x2)) 19.59/7.77 mark(x(x0, x1)) 19.59/7.77 mark(tt) 19.59/7.77 mark(s(x0)) 19.59/7.77 mark(0) 19.59/7.77 a__U11(x0, x1) 19.59/7.77 a__U12(x0) 19.59/7.77 a__isNat(x0) 19.59/7.77 a__U21(x0) 19.59/7.77 a__U31(x0, x1) 19.59/7.77 a__U32(x0) 19.59/7.77 a__U41(x0, x1) 19.59/7.77 a__U51(x0, x1, x2) 19.59/7.77 a__U52(x0, x1, x2) 19.59/7.77 a__plus(x0, x1) 19.59/7.77 a__U61(x0) 19.59/7.77 a__U71(x0, x1, x2) 19.59/7.77 a__U72(x0, x1, x2) 19.59/7.77 a__x(x0, x1) 19.59/7.77 19.59/7.77 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (20) 19.59/7.77 Obligation: 19.59/7.77 Q DP problem: 19.59/7.77 The TRS P consists of the following rules: 19.59/7.77 19.59/7.77 MARK(U12(X)) -> MARK(X) 19.59/7.77 MARK(U11(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U21(X)) -> MARK(X) 19.59/7.77 MARK(U31(X1, X2)) -> MARK(X1) 19.59/7.77 MARK(U32(X)) -> MARK(X) 19.59/7.77 MARK(U61(X)) -> MARK(X) 19.59/7.77 19.59/7.77 R is empty. 19.59/7.77 Q is empty. 19.59/7.77 We have to consider all minimal (P,Q,R)-chains. 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (21) QDPSizeChangeProof (EQUIVALENT) 19.59/7.77 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. 19.59/7.77 19.59/7.77 From the DPs we obtained the following set of size-change graphs: 19.59/7.77 *MARK(U12(X)) -> MARK(X) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 *MARK(U11(X1, X2)) -> MARK(X1) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 *MARK(U21(X)) -> MARK(X) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 *MARK(U31(X1, X2)) -> MARK(X1) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 *MARK(U32(X)) -> MARK(X) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 *MARK(U61(X)) -> MARK(X) 19.59/7.77 The graph contains the following edges 1 > 1 19.59/7.77 19.59/7.77 19.59/7.77 ---------------------------------------- 19.59/7.77 19.59/7.77 (22) 19.59/7.77 YES 19.67/7.88 EOF