11.97/4.22 YES 11.97/4.24 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 11.97/4.24 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.97/4.24 11.97/4.24 11.97/4.24 Termination w.r.t. Q of the given QTRS could be proven: 11.97/4.24 11.97/4.24 (0) QTRS 11.97/4.24 (1) DependencyPairsProof [EQUIVALENT, 136 ms] 11.97/4.24 (2) QDP 11.97/4.24 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 11.97/4.24 (4) AND 11.97/4.24 (5) QDP 11.97/4.24 (6) UsableRulesProof [EQUIVALENT, 0 ms] 11.97/4.24 (7) QDP 11.97/4.24 (8) QReductionProof [EQUIVALENT, 0 ms] 11.97/4.24 (9) QDP 11.97/4.24 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.97/4.24 (11) YES 11.97/4.24 (12) QDP 11.97/4.24 (13) UsableRulesProof [EQUIVALENT, 0 ms] 11.97/4.24 (14) QDP 11.97/4.24 (15) QReductionProof [EQUIVALENT, 0 ms] 11.97/4.24 (16) QDP 11.97/4.24 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.97/4.24 (18) YES 11.97/4.24 (19) QDP 11.97/4.24 (20) QDPOrderProof [EQUIVALENT, 315 ms] 11.97/4.24 (21) QDP 11.97/4.24 (22) DependencyGraphProof [EQUIVALENT, 0 ms] 11.97/4.24 (23) QDP 11.97/4.24 (24) UsableRulesProof [EQUIVALENT, 0 ms] 11.97/4.24 (25) QDP 11.97/4.24 (26) QReductionProof [EQUIVALENT, 0 ms] 11.97/4.24 (27) QDP 11.97/4.24 (28) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.97/4.24 (29) YES 11.97/4.24 11.97/4.24 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (0) 11.97/4.24 Obligation: 11.97/4.24 Q restricted rewrite system: 11.97/4.24 The TRS R consists of the following rules: 11.97/4.24 11.97/4.24 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.24 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.24 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.24 a__U16(tt) -> tt 11.97/4.24 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.24 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.24 a__U23(tt) -> tt 11.97/4.24 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.24 a__U32(tt) -> tt 11.97/4.24 a__U41(tt) -> tt 11.97/4.24 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.24 a__U52(tt, N) -> mark(N) 11.97/4.24 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.24 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.24 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.24 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.24 a__isNat(0) -> tt 11.97/4.24 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.24 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.24 a__isNatKind(0) -> tt 11.97/4.24 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.24 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.24 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.24 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.24 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.24 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.24 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.24 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.24 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.24 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.24 mark(isNat(X)) -> a__isNat(X) 11.97/4.24 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.24 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.24 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.24 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.24 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.24 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.24 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.24 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.24 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.24 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.24 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.24 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.24 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.24 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.24 mark(tt) -> tt 11.97/4.24 mark(s(X)) -> s(mark(X)) 11.97/4.24 mark(0) -> 0 11.97/4.24 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.24 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.24 a__isNatKind(X) -> isNatKind(X) 11.97/4.24 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.24 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.24 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.24 a__isNat(X) -> isNat(X) 11.97/4.24 a__U16(X) -> U16(X) 11.97/4.24 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.24 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.24 a__U23(X) -> U23(X) 11.97/4.24 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.24 a__U32(X) -> U32(X) 11.97/4.24 a__U41(X) -> U41(X) 11.97/4.24 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.24 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.24 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.24 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.24 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.24 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.24 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.24 11.97/4.24 The set Q consists of the following terms: 11.97/4.24 11.97/4.24 mark(U11(x0, x1, x2)) 11.97/4.24 mark(U12(x0, x1, x2)) 11.97/4.24 mark(isNatKind(x0)) 11.97/4.24 mark(U13(x0, x1, x2)) 11.97/4.24 mark(U14(x0, x1, x2)) 11.97/4.24 mark(U15(x0, x1)) 11.97/4.24 mark(isNat(x0)) 11.97/4.24 mark(U16(x0)) 11.97/4.24 mark(U21(x0, x1)) 11.97/4.24 mark(U22(x0, x1)) 11.97/4.24 mark(U23(x0)) 11.97/4.24 mark(U31(x0, x1)) 11.97/4.24 mark(U32(x0)) 11.97/4.24 mark(U41(x0)) 11.97/4.24 mark(U51(x0, x1)) 11.97/4.24 mark(U52(x0, x1)) 11.97/4.24 mark(U61(x0, x1, x2)) 11.97/4.24 mark(U62(x0, x1, x2)) 11.97/4.24 mark(U63(x0, x1, x2)) 11.97/4.24 mark(U64(x0, x1, x2)) 11.97/4.24 mark(plus(x0, x1)) 11.97/4.24 mark(tt) 11.97/4.24 mark(s(x0)) 11.97/4.24 mark(0) 11.97/4.24 a__U11(x0, x1, x2) 11.97/4.24 a__U12(x0, x1, x2) 11.97/4.24 a__isNatKind(x0) 11.97/4.24 a__U13(x0, x1, x2) 11.97/4.24 a__U14(x0, x1, x2) 11.97/4.24 a__U15(x0, x1) 11.97/4.24 a__isNat(x0) 11.97/4.24 a__U16(x0) 11.97/4.24 a__U21(x0, x1) 11.97/4.24 a__U22(x0, x1) 11.97/4.24 a__U23(x0) 11.97/4.24 a__U31(x0, x1) 11.97/4.24 a__U32(x0) 11.97/4.24 a__U41(x0) 11.97/4.24 a__U51(x0, x1) 11.97/4.24 a__U52(x0, x1) 11.97/4.24 a__U61(x0, x1, x2) 11.97/4.24 a__U62(x0, x1, x2) 11.97/4.24 a__U63(x0, x1, x2) 11.97/4.24 a__U64(x0, x1, x2) 11.97/4.24 a__plus(x0, x1) 11.97/4.24 11.97/4.24 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (1) DependencyPairsProof (EQUIVALENT) 11.97/4.24 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (2) 11.97/4.24 Obligation: 11.97/4.24 Q DP problem: 11.97/4.24 The TRS P consists of the following rules: 11.97/4.24 11.97/4.24 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 11.97/4.24 A__U11(tt, V1, V2) -> A__ISNATKIND(V1) 11.97/4.24 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 11.97/4.24 A__U12(tt, V1, V2) -> A__ISNATKIND(V2) 11.97/4.24 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 11.97/4.24 A__U13(tt, V1, V2) -> A__ISNATKIND(V2) 11.97/4.24 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 11.97/4.24 A__U14(tt, V1, V2) -> A__ISNAT(V1) 11.97/4.24 A__U15(tt, V2) -> A__U16(a__isNat(V2)) 11.97/4.24 A__U15(tt, V2) -> A__ISNAT(V2) 11.97/4.24 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 11.97/4.24 A__U21(tt, V1) -> A__ISNATKIND(V1) 11.97/4.24 A__U22(tt, V1) -> A__U23(a__isNat(V1)) 11.97/4.24 A__U22(tt, V1) -> A__ISNAT(V1) 11.97/4.24 A__U31(tt, V2) -> A__U32(a__isNatKind(V2)) 11.97/4.24 A__U31(tt, V2) -> A__ISNATKIND(V2) 11.97/4.24 A__U51(tt, N) -> A__U52(a__isNatKind(N), N) 11.97/4.24 A__U51(tt, N) -> A__ISNATKIND(N) 11.97/4.24 A__U52(tt, N) -> MARK(N) 11.97/4.24 A__U61(tt, M, N) -> A__U62(a__isNatKind(M), M, N) 11.97/4.24 A__U61(tt, M, N) -> A__ISNATKIND(M) 11.97/4.24 A__U62(tt, M, N) -> A__U63(a__isNat(N), M, N) 11.97/4.24 A__U62(tt, M, N) -> A__ISNAT(N) 11.97/4.24 A__U63(tt, M, N) -> A__U64(a__isNatKind(N), M, N) 11.97/4.24 A__U63(tt, M, N) -> A__ISNATKIND(N) 11.97/4.24 A__U64(tt, M, N) -> A__PLUS(mark(N), mark(M)) 11.97/4.24 A__U64(tt, M, N) -> MARK(N) 11.97/4.24 A__U64(tt, M, N) -> MARK(M) 11.97/4.24 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 11.97/4.24 A__ISNAT(plus(V1, V2)) -> A__ISNATKIND(V1) 11.97/4.24 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 11.97/4.24 A__ISNAT(s(V1)) -> A__ISNATKIND(V1) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__U31(a__isNatKind(V1), V2) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 11.97/4.24 A__ISNATKIND(s(V1)) -> A__U41(a__isNatKind(V1)) 11.97/4.24 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 11.97/4.24 A__PLUS(N, 0) -> A__U51(a__isNat(N), N) 11.97/4.24 A__PLUS(N, 0) -> A__ISNAT(N) 11.97/4.24 A__PLUS(N, s(M)) -> A__U61(a__isNat(M), M, N) 11.97/4.24 A__PLUS(N, s(M)) -> A__ISNAT(M) 11.97/4.24 MARK(U11(X1, X2, X3)) -> A__U11(mark(X1), X2, X3) 11.97/4.24 MARK(U11(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(U12(X1, X2, X3)) -> A__U12(mark(X1), X2, X3) 11.97/4.24 MARK(U12(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(isNatKind(X)) -> A__ISNATKIND(X) 11.97/4.24 MARK(U13(X1, X2, X3)) -> A__U13(mark(X1), X2, X3) 11.97/4.24 MARK(U13(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(U14(X1, X2, X3)) -> A__U14(mark(X1), X2, X3) 11.97/4.24 MARK(U14(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(U15(X1, X2)) -> A__U15(mark(X1), X2) 11.97/4.24 MARK(U15(X1, X2)) -> MARK(X1) 11.97/4.24 MARK(isNat(X)) -> A__ISNAT(X) 11.97/4.24 MARK(U16(X)) -> A__U16(mark(X)) 11.97/4.24 MARK(U16(X)) -> MARK(X) 11.97/4.24 MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) 11.97/4.24 MARK(U21(X1, X2)) -> MARK(X1) 11.97/4.24 MARK(U22(X1, X2)) -> A__U22(mark(X1), X2) 11.97/4.24 MARK(U22(X1, X2)) -> MARK(X1) 11.97/4.24 MARK(U23(X)) -> A__U23(mark(X)) 11.97/4.24 MARK(U23(X)) -> MARK(X) 11.97/4.24 MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) 11.97/4.24 MARK(U31(X1, X2)) -> MARK(X1) 11.97/4.24 MARK(U32(X)) -> A__U32(mark(X)) 11.97/4.24 MARK(U32(X)) -> MARK(X) 11.97/4.24 MARK(U41(X)) -> A__U41(mark(X)) 11.97/4.24 MARK(U41(X)) -> MARK(X) 11.97/4.24 MARK(U51(X1, X2)) -> A__U51(mark(X1), X2) 11.97/4.24 MARK(U51(X1, X2)) -> MARK(X1) 11.97/4.24 MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) 11.97/4.24 MARK(U52(X1, X2)) -> MARK(X1) 11.97/4.24 MARK(U61(X1, X2, X3)) -> A__U61(mark(X1), X2, X3) 11.97/4.24 MARK(U61(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(U62(X1, X2, X3)) -> A__U62(mark(X1), X2, X3) 11.97/4.24 MARK(U62(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(U63(X1, X2, X3)) -> A__U63(mark(X1), X2, X3) 11.97/4.24 MARK(U63(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(U64(X1, X2, X3)) -> A__U64(mark(X1), X2, X3) 11.97/4.24 MARK(U64(X1, X2, X3)) -> MARK(X1) 11.97/4.24 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 11.97/4.24 MARK(plus(X1, X2)) -> MARK(X1) 11.97/4.24 MARK(plus(X1, X2)) -> MARK(X2) 11.97/4.24 MARK(s(X)) -> MARK(X) 11.97/4.24 11.97/4.24 The TRS R consists of the following rules: 11.97/4.24 11.97/4.24 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.24 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.24 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.24 a__U16(tt) -> tt 11.97/4.24 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.24 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.24 a__U23(tt) -> tt 11.97/4.24 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.24 a__U32(tt) -> tt 11.97/4.24 a__U41(tt) -> tt 11.97/4.24 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.24 a__U52(tt, N) -> mark(N) 11.97/4.24 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.24 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.24 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.24 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.24 a__isNat(0) -> tt 11.97/4.24 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.24 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.24 a__isNatKind(0) -> tt 11.97/4.24 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.24 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.24 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.24 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.24 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.24 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.24 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.24 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.24 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.24 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.24 mark(isNat(X)) -> a__isNat(X) 11.97/4.24 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.24 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.24 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.24 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.24 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.24 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.24 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.24 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.24 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.24 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.24 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.24 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.24 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.24 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.24 mark(tt) -> tt 11.97/4.24 mark(s(X)) -> s(mark(X)) 11.97/4.24 mark(0) -> 0 11.97/4.24 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.24 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.24 a__isNatKind(X) -> isNatKind(X) 11.97/4.24 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.24 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.24 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.24 a__isNat(X) -> isNat(X) 11.97/4.24 a__U16(X) -> U16(X) 11.97/4.24 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.24 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.24 a__U23(X) -> U23(X) 11.97/4.24 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.24 a__U32(X) -> U32(X) 11.97/4.24 a__U41(X) -> U41(X) 11.97/4.24 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.24 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.24 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.24 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.24 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.24 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.24 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.24 11.97/4.24 The set Q consists of the following terms: 11.97/4.24 11.97/4.24 mark(U11(x0, x1, x2)) 11.97/4.24 mark(U12(x0, x1, x2)) 11.97/4.24 mark(isNatKind(x0)) 11.97/4.24 mark(U13(x0, x1, x2)) 11.97/4.24 mark(U14(x0, x1, x2)) 11.97/4.24 mark(U15(x0, x1)) 11.97/4.24 mark(isNat(x0)) 11.97/4.24 mark(U16(x0)) 11.97/4.24 mark(U21(x0, x1)) 11.97/4.24 mark(U22(x0, x1)) 11.97/4.24 mark(U23(x0)) 11.97/4.24 mark(U31(x0, x1)) 11.97/4.24 mark(U32(x0)) 11.97/4.24 mark(U41(x0)) 11.97/4.24 mark(U51(x0, x1)) 11.97/4.24 mark(U52(x0, x1)) 11.97/4.24 mark(U61(x0, x1, x2)) 11.97/4.24 mark(U62(x0, x1, x2)) 11.97/4.24 mark(U63(x0, x1, x2)) 11.97/4.24 mark(U64(x0, x1, x2)) 11.97/4.24 mark(plus(x0, x1)) 11.97/4.24 mark(tt) 11.97/4.24 mark(s(x0)) 11.97/4.24 mark(0) 11.97/4.24 a__U11(x0, x1, x2) 11.97/4.24 a__U12(x0, x1, x2) 11.97/4.24 a__isNatKind(x0) 11.97/4.24 a__U13(x0, x1, x2) 11.97/4.24 a__U14(x0, x1, x2) 11.97/4.24 a__U15(x0, x1) 11.97/4.24 a__isNat(x0) 11.97/4.24 a__U16(x0) 11.97/4.24 a__U21(x0, x1) 11.97/4.24 a__U22(x0, x1) 11.97/4.24 a__U23(x0) 11.97/4.24 a__U31(x0, x1) 11.97/4.24 a__U32(x0) 11.97/4.24 a__U41(x0) 11.97/4.24 a__U51(x0, x1) 11.97/4.24 a__U52(x0, x1) 11.97/4.24 a__U61(x0, x1, x2) 11.97/4.24 a__U62(x0, x1, x2) 11.97/4.24 a__U63(x0, x1, x2) 11.97/4.24 a__U64(x0, x1, x2) 11.97/4.24 a__plus(x0, x1) 11.97/4.24 11.97/4.24 We have to consider all minimal (P,Q,R)-chains. 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (3) DependencyGraphProof (EQUIVALENT) 11.97/4.24 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 30 less nodes. 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (4) 11.97/4.24 Complex Obligation (AND) 11.97/4.24 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (5) 11.97/4.24 Obligation: 11.97/4.24 Q DP problem: 11.97/4.24 The TRS P consists of the following rules: 11.97/4.24 11.97/4.24 A__U31(tt, V2) -> A__ISNATKIND(V2) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__U31(a__isNatKind(V1), V2) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 11.97/4.24 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 11.97/4.24 11.97/4.24 The TRS R consists of the following rules: 11.97/4.24 11.97/4.24 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.24 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.24 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.24 a__U16(tt) -> tt 11.97/4.24 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.24 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.24 a__U23(tt) -> tt 11.97/4.24 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.24 a__U32(tt) -> tt 11.97/4.24 a__U41(tt) -> tt 11.97/4.24 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.24 a__U52(tt, N) -> mark(N) 11.97/4.24 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.24 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.24 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.24 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.24 a__isNat(0) -> tt 11.97/4.24 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.24 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.24 a__isNatKind(0) -> tt 11.97/4.24 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.24 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.24 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.24 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.24 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.24 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.24 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.24 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.24 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.24 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.24 mark(isNat(X)) -> a__isNat(X) 11.97/4.24 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.24 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.24 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.24 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.24 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.24 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.24 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.24 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.24 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.24 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.24 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.24 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.24 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.24 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.24 mark(tt) -> tt 11.97/4.24 mark(s(X)) -> s(mark(X)) 11.97/4.24 mark(0) -> 0 11.97/4.24 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.24 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.24 a__isNatKind(X) -> isNatKind(X) 11.97/4.24 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.24 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.24 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.24 a__isNat(X) -> isNat(X) 11.97/4.24 a__U16(X) -> U16(X) 11.97/4.24 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.24 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.24 a__U23(X) -> U23(X) 11.97/4.24 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.24 a__U32(X) -> U32(X) 11.97/4.24 a__U41(X) -> U41(X) 11.97/4.24 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.24 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.24 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.24 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.24 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.24 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.24 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.24 11.97/4.24 The set Q consists of the following terms: 11.97/4.24 11.97/4.24 mark(U11(x0, x1, x2)) 11.97/4.24 mark(U12(x0, x1, x2)) 11.97/4.24 mark(isNatKind(x0)) 11.97/4.24 mark(U13(x0, x1, x2)) 11.97/4.24 mark(U14(x0, x1, x2)) 11.97/4.24 mark(U15(x0, x1)) 11.97/4.24 mark(isNat(x0)) 11.97/4.24 mark(U16(x0)) 11.97/4.24 mark(U21(x0, x1)) 11.97/4.24 mark(U22(x0, x1)) 11.97/4.24 mark(U23(x0)) 11.97/4.24 mark(U31(x0, x1)) 11.97/4.24 mark(U32(x0)) 11.97/4.24 mark(U41(x0)) 11.97/4.24 mark(U51(x0, x1)) 11.97/4.24 mark(U52(x0, x1)) 11.97/4.24 mark(U61(x0, x1, x2)) 11.97/4.24 mark(U62(x0, x1, x2)) 11.97/4.24 mark(U63(x0, x1, x2)) 11.97/4.24 mark(U64(x0, x1, x2)) 11.97/4.24 mark(plus(x0, x1)) 11.97/4.24 mark(tt) 11.97/4.24 mark(s(x0)) 11.97/4.24 mark(0) 11.97/4.24 a__U11(x0, x1, x2) 11.97/4.24 a__U12(x0, x1, x2) 11.97/4.24 a__isNatKind(x0) 11.97/4.24 a__U13(x0, x1, x2) 11.97/4.24 a__U14(x0, x1, x2) 11.97/4.24 a__U15(x0, x1) 11.97/4.24 a__isNat(x0) 11.97/4.24 a__U16(x0) 11.97/4.24 a__U21(x0, x1) 11.97/4.24 a__U22(x0, x1) 11.97/4.24 a__U23(x0) 11.97/4.24 a__U31(x0, x1) 11.97/4.24 a__U32(x0) 11.97/4.24 a__U41(x0) 11.97/4.24 a__U51(x0, x1) 11.97/4.24 a__U52(x0, x1) 11.97/4.24 a__U61(x0, x1, x2) 11.97/4.24 a__U62(x0, x1, x2) 11.97/4.24 a__U63(x0, x1, x2) 11.97/4.24 a__U64(x0, x1, x2) 11.97/4.24 a__plus(x0, x1) 11.97/4.24 11.97/4.24 We have to consider all minimal (P,Q,R)-chains. 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (6) UsableRulesProof (EQUIVALENT) 11.97/4.24 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. 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (7) 11.97/4.24 Obligation: 11.97/4.24 Q DP problem: 11.97/4.24 The TRS P consists of the following rules: 11.97/4.24 11.97/4.24 A__U31(tt, V2) -> A__ISNATKIND(V2) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__U31(a__isNatKind(V1), V2) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 11.97/4.24 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 11.97/4.24 11.97/4.24 The TRS R consists of the following rules: 11.97/4.24 11.97/4.24 a__isNatKind(0) -> tt 11.97/4.24 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.24 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.24 a__isNatKind(X) -> isNatKind(X) 11.97/4.24 a__U41(tt) -> tt 11.97/4.24 a__U41(X) -> U41(X) 11.97/4.24 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.24 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.24 a__U32(tt) -> tt 11.97/4.24 a__U32(X) -> U32(X) 11.97/4.24 11.97/4.24 The set Q consists of the following terms: 11.97/4.24 11.97/4.24 mark(U11(x0, x1, x2)) 11.97/4.24 mark(U12(x0, x1, x2)) 11.97/4.24 mark(isNatKind(x0)) 11.97/4.24 mark(U13(x0, x1, x2)) 11.97/4.24 mark(U14(x0, x1, x2)) 11.97/4.24 mark(U15(x0, x1)) 11.97/4.24 mark(isNat(x0)) 11.97/4.24 mark(U16(x0)) 11.97/4.24 mark(U21(x0, x1)) 11.97/4.24 mark(U22(x0, x1)) 11.97/4.24 mark(U23(x0)) 11.97/4.24 mark(U31(x0, x1)) 11.97/4.24 mark(U32(x0)) 11.97/4.24 mark(U41(x0)) 11.97/4.24 mark(U51(x0, x1)) 11.97/4.24 mark(U52(x0, x1)) 11.97/4.24 mark(U61(x0, x1, x2)) 11.97/4.24 mark(U62(x0, x1, x2)) 11.97/4.24 mark(U63(x0, x1, x2)) 11.97/4.24 mark(U64(x0, x1, x2)) 11.97/4.24 mark(plus(x0, x1)) 11.97/4.24 mark(tt) 11.97/4.24 mark(s(x0)) 11.97/4.24 mark(0) 11.97/4.24 a__U11(x0, x1, x2) 11.97/4.24 a__U12(x0, x1, x2) 11.97/4.24 a__isNatKind(x0) 11.97/4.24 a__U13(x0, x1, x2) 11.97/4.24 a__U14(x0, x1, x2) 11.97/4.24 a__U15(x0, x1) 11.97/4.24 a__isNat(x0) 11.97/4.24 a__U16(x0) 11.97/4.24 a__U21(x0, x1) 11.97/4.24 a__U22(x0, x1) 11.97/4.24 a__U23(x0) 11.97/4.24 a__U31(x0, x1) 11.97/4.24 a__U32(x0) 11.97/4.24 a__U41(x0) 11.97/4.24 a__U51(x0, x1) 11.97/4.24 a__U52(x0, x1) 11.97/4.24 a__U61(x0, x1, x2) 11.97/4.24 a__U62(x0, x1, x2) 11.97/4.24 a__U63(x0, x1, x2) 11.97/4.24 a__U64(x0, x1, x2) 11.97/4.24 a__plus(x0, x1) 11.97/4.24 11.97/4.24 We have to consider all minimal (P,Q,R)-chains. 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (8) QReductionProof (EQUIVALENT) 11.97/4.24 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 11.97/4.24 11.97/4.24 mark(U11(x0, x1, x2)) 11.97/4.24 mark(U12(x0, x1, x2)) 11.97/4.24 mark(isNatKind(x0)) 11.97/4.24 mark(U13(x0, x1, x2)) 11.97/4.24 mark(U14(x0, x1, x2)) 11.97/4.24 mark(U15(x0, x1)) 11.97/4.24 mark(isNat(x0)) 11.97/4.24 mark(U16(x0)) 11.97/4.24 mark(U21(x0, x1)) 11.97/4.24 mark(U22(x0, x1)) 11.97/4.24 mark(U23(x0)) 11.97/4.24 mark(U31(x0, x1)) 11.97/4.24 mark(U32(x0)) 11.97/4.24 mark(U41(x0)) 11.97/4.24 mark(U51(x0, x1)) 11.97/4.24 mark(U52(x0, x1)) 11.97/4.24 mark(U61(x0, x1, x2)) 11.97/4.24 mark(U62(x0, x1, x2)) 11.97/4.24 mark(U63(x0, x1, x2)) 11.97/4.24 mark(U64(x0, x1, x2)) 11.97/4.24 mark(plus(x0, x1)) 11.97/4.24 mark(tt) 11.97/4.24 mark(s(x0)) 11.97/4.24 mark(0) 11.97/4.24 a__U11(x0, x1, x2) 11.97/4.24 a__U12(x0, x1, x2) 11.97/4.24 a__U13(x0, x1, x2) 11.97/4.24 a__U14(x0, x1, x2) 11.97/4.24 a__U15(x0, x1) 11.97/4.24 a__isNat(x0) 11.97/4.24 a__U16(x0) 11.97/4.24 a__U21(x0, x1) 11.97/4.24 a__U22(x0, x1) 11.97/4.24 a__U23(x0) 11.97/4.24 a__U51(x0, x1) 11.97/4.24 a__U52(x0, x1) 11.97/4.24 a__U61(x0, x1, x2) 11.97/4.24 a__U62(x0, x1, x2) 11.97/4.24 a__U63(x0, x1, x2) 11.97/4.24 a__U64(x0, x1, x2) 11.97/4.24 a__plus(x0, x1) 11.97/4.24 11.97/4.24 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (9) 11.97/4.24 Obligation: 11.97/4.24 Q DP problem: 11.97/4.24 The TRS P consists of the following rules: 11.97/4.24 11.97/4.24 A__U31(tt, V2) -> A__ISNATKIND(V2) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__U31(a__isNatKind(V1), V2) 11.97/4.24 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 11.97/4.24 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 11.97/4.24 11.97/4.24 The TRS R consists of the following rules: 11.97/4.24 11.97/4.24 a__isNatKind(0) -> tt 11.97/4.24 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.24 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.24 a__isNatKind(X) -> isNatKind(X) 11.97/4.24 a__U41(tt) -> tt 11.97/4.24 a__U41(X) -> U41(X) 11.97/4.24 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.24 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.24 a__U32(tt) -> tt 11.97/4.24 a__U32(X) -> U32(X) 11.97/4.24 11.97/4.24 The set Q consists of the following terms: 11.97/4.24 11.97/4.24 a__isNatKind(x0) 11.97/4.24 a__U31(x0, x1) 11.97/4.24 a__U32(x0) 11.97/4.24 a__U41(x0) 11.97/4.24 11.97/4.24 We have to consider all minimal (P,Q,R)-chains. 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (10) QDPSizeChangeProof (EQUIVALENT) 11.97/4.24 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. 11.97/4.24 11.97/4.24 From the DPs we obtained the following set of size-change graphs: 11.97/4.24 *A__ISNATKIND(plus(V1, V2)) -> A__U31(a__isNatKind(V1), V2) 11.97/4.24 The graph contains the following edges 1 > 2 11.97/4.24 11.97/4.24 11.97/4.24 *A__U31(tt, V2) -> A__ISNATKIND(V2) 11.97/4.24 The graph contains the following edges 2 >= 1 11.97/4.24 11.97/4.24 11.97/4.24 *A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 11.97/4.24 The graph contains the following edges 1 > 1 11.97/4.24 11.97/4.24 11.97/4.24 *A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 11.97/4.24 The graph contains the following edges 1 > 1 11.97/4.24 11.97/4.24 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (11) 11.97/4.24 YES 11.97/4.24 11.97/4.24 ---------------------------------------- 11.97/4.24 11.97/4.24 (12) 11.97/4.24 Obligation: 11.97/4.24 Q DP problem: 11.97/4.24 The TRS P consists of the following rules: 11.97/4.24 11.97/4.24 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 11.97/4.24 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 11.97/4.24 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 11.97/4.24 A__U15(tt, V2) -> A__ISNAT(V2) 11.97/4.24 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 11.97/4.24 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 11.97/4.24 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 11.97/4.24 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 11.97/4.24 A__U22(tt, V1) -> A__ISNAT(V1) 11.97/4.24 A__U14(tt, V1, V2) -> A__ISNAT(V1) 11.97/4.24 11.97/4.24 The TRS R consists of the following rules: 11.97/4.24 11.97/4.24 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.24 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.24 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.24 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.24 a__U16(tt) -> tt 11.97/4.24 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.24 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.24 a__U23(tt) -> tt 11.97/4.24 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.24 a__U32(tt) -> tt 11.97/4.24 a__U41(tt) -> tt 11.97/4.24 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.24 a__U52(tt, N) -> mark(N) 11.97/4.24 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.24 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.24 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.24 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.24 a__isNat(0) -> tt 11.97/4.24 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.24 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.24 a__isNatKind(0) -> tt 11.97/4.24 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.24 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.24 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.24 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.24 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.24 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.24 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.24 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.24 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.24 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.24 mark(isNat(X)) -> a__isNat(X) 11.97/4.24 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.24 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.24 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.24 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.24 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.24 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.24 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.24 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.24 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.24 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.24 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.24 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.24 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.24 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.24 mark(tt) -> tt 11.97/4.24 mark(s(X)) -> s(mark(X)) 11.97/4.24 mark(0) -> 0 11.97/4.24 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.24 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.24 a__isNatKind(X) -> isNatKind(X) 11.97/4.24 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.25 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.25 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.25 a__isNat(X) -> isNat(X) 11.97/4.25 a__U16(X) -> U16(X) 11.97/4.25 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.25 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.25 a__U23(X) -> U23(X) 11.97/4.25 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.25 a__U32(X) -> U32(X) 11.97/4.25 a__U41(X) -> U41(X) 11.97/4.25 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.25 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.25 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.25 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.25 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.25 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.25 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.25 11.97/4.25 The set Q consists of the following terms: 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (13) UsableRulesProof (EQUIVALENT) 11.97/4.25 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. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (14) 11.97/4.25 Obligation: 11.97/4.25 Q DP problem: 11.97/4.25 The TRS P consists of the following rules: 11.97/4.25 11.97/4.25 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 11.97/4.25 A__U15(tt, V2) -> A__ISNAT(V2) 11.97/4.25 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 11.97/4.25 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 11.97/4.25 A__U22(tt, V1) -> A__ISNAT(V1) 11.97/4.25 A__U14(tt, V1, V2) -> A__ISNAT(V1) 11.97/4.25 11.97/4.25 The TRS R consists of the following rules: 11.97/4.25 11.97/4.25 a__isNatKind(0) -> tt 11.97/4.25 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.25 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.25 a__isNatKind(X) -> isNatKind(X) 11.97/4.25 a__U41(tt) -> tt 11.97/4.25 a__U41(X) -> U41(X) 11.97/4.25 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.25 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.25 a__U32(tt) -> tt 11.97/4.25 a__U32(X) -> U32(X) 11.97/4.25 a__isNat(0) -> tt 11.97/4.25 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.25 a__isNat(X) -> isNat(X) 11.97/4.25 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.25 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.25 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.25 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.25 a__U23(tt) -> tt 11.97/4.25 a__U23(X) -> U23(X) 11.97/4.25 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.25 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.25 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.25 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.25 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.25 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.25 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.25 a__U16(tt) -> tt 11.97/4.25 a__U16(X) -> U16(X) 11.97/4.25 11.97/4.25 The set Q consists of the following terms: 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (15) QReductionProof (EQUIVALENT) 11.97/4.25 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (16) 11.97/4.25 Obligation: 11.97/4.25 Q DP problem: 11.97/4.25 The TRS P consists of the following rules: 11.97/4.25 11.97/4.25 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 11.97/4.25 A__U15(tt, V2) -> A__ISNAT(V2) 11.97/4.25 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 11.97/4.25 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 11.97/4.25 A__U22(tt, V1) -> A__ISNAT(V1) 11.97/4.25 A__U14(tt, V1, V2) -> A__ISNAT(V1) 11.97/4.25 11.97/4.25 The TRS R consists of the following rules: 11.97/4.25 11.97/4.25 a__isNatKind(0) -> tt 11.97/4.25 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.25 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.25 a__isNatKind(X) -> isNatKind(X) 11.97/4.25 a__U41(tt) -> tt 11.97/4.25 a__U41(X) -> U41(X) 11.97/4.25 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.25 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.25 a__U32(tt) -> tt 11.97/4.25 a__U32(X) -> U32(X) 11.97/4.25 a__isNat(0) -> tt 11.97/4.25 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.25 a__isNat(X) -> isNat(X) 11.97/4.25 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.25 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.25 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.25 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.25 a__U23(tt) -> tt 11.97/4.25 a__U23(X) -> U23(X) 11.97/4.25 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.25 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.25 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.25 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.25 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.25 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.25 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.25 a__U16(tt) -> tt 11.97/4.25 a__U16(X) -> U16(X) 11.97/4.25 11.97/4.25 The set Q consists of the following terms: 11.97/4.25 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (17) QDPSizeChangeProof (EQUIVALENT) 11.97/4.25 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. 11.97/4.25 11.97/4.25 From the DPs we obtained the following set of size-change graphs: 11.97/4.25 *A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 The graph contains the following edges 2 >= 2, 3 >= 3 11.97/4.25 11.97/4.25 11.97/4.25 *A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 The graph contains the following edges 2 >= 2, 3 >= 3 11.97/4.25 11.97/4.25 11.97/4.25 *A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 The graph contains the following edges 2 >= 2, 3 >= 3 11.97/4.25 11.97/4.25 11.97/4.25 *A__U15(tt, V2) -> A__ISNAT(V2) 11.97/4.25 The graph contains the following edges 2 >= 1 11.97/4.25 11.97/4.25 11.97/4.25 *A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 11.97/4.25 The graph contains the following edges 3 >= 2 11.97/4.25 11.97/4.25 11.97/4.25 *A__U14(tt, V1, V2) -> A__ISNAT(V1) 11.97/4.25 The graph contains the following edges 2 >= 1 11.97/4.25 11.97/4.25 11.97/4.25 *A__U22(tt, V1) -> A__ISNAT(V1) 11.97/4.25 The graph contains the following edges 2 >= 1 11.97/4.25 11.97/4.25 11.97/4.25 *A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 11.97/4.25 The graph contains the following edges 2 >= 2 11.97/4.25 11.97/4.25 11.97/4.25 *A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 The graph contains the following edges 1 > 2, 1 > 3 11.97/4.25 11.97/4.25 11.97/4.25 *A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 11.97/4.25 The graph contains the following edges 1 > 2 11.97/4.25 11.97/4.25 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (18) 11.97/4.25 YES 11.97/4.25 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (19) 11.97/4.25 Obligation: 11.97/4.25 Q DP problem: 11.97/4.25 The TRS P consists of the following rules: 11.97/4.25 11.97/4.25 MARK(U11(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U12(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U13(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U14(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U15(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U16(X)) -> MARK(X) 11.97/4.25 MARK(U21(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U22(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U23(X)) -> MARK(X) 11.97/4.25 MARK(U31(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U32(X)) -> MARK(X) 11.97/4.25 MARK(U41(X)) -> MARK(X) 11.97/4.25 MARK(U51(X1, X2)) -> A__U51(mark(X1), X2) 11.97/4.25 A__U51(tt, N) -> A__U52(a__isNatKind(N), N) 11.97/4.25 A__U52(tt, N) -> MARK(N) 11.97/4.25 MARK(U51(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) 11.97/4.25 MARK(U52(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U61(X1, X2, X3)) -> A__U61(mark(X1), X2, X3) 11.97/4.25 A__U61(tt, M, N) -> A__U62(a__isNatKind(M), M, N) 11.97/4.25 A__U62(tt, M, N) -> A__U63(a__isNat(N), M, N) 11.97/4.25 A__U63(tt, M, N) -> A__U64(a__isNatKind(N), M, N) 11.97/4.25 A__U64(tt, M, N) -> A__PLUS(mark(N), mark(M)) 11.97/4.25 A__PLUS(N, 0) -> A__U51(a__isNat(N), N) 11.97/4.25 A__PLUS(N, s(M)) -> A__U61(a__isNat(M), M, N) 11.97/4.25 A__U64(tt, M, N) -> MARK(N) 11.97/4.25 MARK(U61(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U62(X1, X2, X3)) -> A__U62(mark(X1), X2, X3) 11.97/4.25 MARK(U62(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U63(X1, X2, X3)) -> A__U63(mark(X1), X2, X3) 11.97/4.25 MARK(U63(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U64(X1, X2, X3)) -> A__U64(mark(X1), X2, X3) 11.97/4.25 A__U64(tt, M, N) -> MARK(M) 11.97/4.25 MARK(U64(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X2) 11.97/4.25 MARK(s(X)) -> MARK(X) 11.97/4.25 11.97/4.25 The TRS R consists of the following rules: 11.97/4.25 11.97/4.25 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.25 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.25 a__U16(tt) -> tt 11.97/4.25 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.25 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.25 a__U23(tt) -> tt 11.97/4.25 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.25 a__U32(tt) -> tt 11.97/4.25 a__U41(tt) -> tt 11.97/4.25 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.25 a__U52(tt, N) -> mark(N) 11.97/4.25 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.25 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.25 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.25 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.25 a__isNat(0) -> tt 11.97/4.25 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.25 a__isNatKind(0) -> tt 11.97/4.25 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.25 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.25 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.25 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.25 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.25 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.25 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.25 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.25 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.25 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.25 mark(isNat(X)) -> a__isNat(X) 11.97/4.25 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.25 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.25 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.25 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.25 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.25 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.25 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.25 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.25 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.25 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.25 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.25 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.25 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.25 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.25 mark(tt) -> tt 11.97/4.25 mark(s(X)) -> s(mark(X)) 11.97/4.25 mark(0) -> 0 11.97/4.25 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.25 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.25 a__isNatKind(X) -> isNatKind(X) 11.97/4.25 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.25 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.25 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.25 a__isNat(X) -> isNat(X) 11.97/4.25 a__U16(X) -> U16(X) 11.97/4.25 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.25 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.25 a__U23(X) -> U23(X) 11.97/4.25 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.25 a__U32(X) -> U32(X) 11.97/4.25 a__U41(X) -> U41(X) 11.97/4.25 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.25 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.25 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.25 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.25 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.25 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.25 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.25 11.97/4.25 The set Q consists of the following terms: 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (20) QDPOrderProof (EQUIVALENT) 11.97/4.25 We use the reduction pair processor [LPAR04,JAR06]. 11.97/4.25 11.97/4.25 11.97/4.25 The following pairs can be oriented strictly and are deleted. 11.97/4.25 11.97/4.25 MARK(U51(X1, X2)) -> A__U51(mark(X1), X2) 11.97/4.25 A__U52(tt, N) -> MARK(N) 11.97/4.25 MARK(U51(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) 11.97/4.25 MARK(U52(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U61(X1, X2, X3)) -> A__U61(mark(X1), X2, X3) 11.97/4.25 A__U63(tt, M, N) -> A__U64(a__isNatKind(N), M, N) 11.97/4.25 A__PLUS(N, 0) -> A__U51(a__isNat(N), N) 11.97/4.25 A__PLUS(N, s(M)) -> A__U61(a__isNat(M), M, N) 11.97/4.25 MARK(U61(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U62(X1, X2, X3)) -> A__U62(mark(X1), X2, X3) 11.97/4.25 MARK(U62(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U63(X1, X2, X3)) -> A__U63(mark(X1), X2, X3) 11.97/4.25 MARK(U63(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U64(X1, X2, X3)) -> A__U64(mark(X1), X2, X3) 11.97/4.25 MARK(U64(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(s(X)) -> MARK(X) 11.97/4.25 The remaining pairs can at least be oriented weakly. 11.97/4.25 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 11.97/4.25 11.97/4.25 POL( A__PLUS_2(x_1, x_2) ) = 2x_1 + 2x_2 11.97/4.25 POL( A__U51_2(x_1, x_2) ) = 2x_2 + 1 11.97/4.25 POL( A__U52_2(x_1, x_2) ) = 2x_2 + 1 11.97/4.25 POL( A__U61_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 11.97/4.25 POL( A__U62_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 11.97/4.25 POL( A__U63_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 11.97/4.25 POL( A__U64_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 11.97/4.25 POL( mark_1(x_1) ) = x_1 11.97/4.25 POL( U11_3(x_1, ..., x_3) ) = 2x_1 11.97/4.25 POL( a__U11_3(x_1, ..., x_3) ) = 2x_1 11.97/4.25 POL( U12_3(x_1, ..., x_3) ) = 2x_1 11.97/4.25 POL( a__U12_3(x_1, ..., x_3) ) = 2x_1 11.97/4.25 POL( isNatKind_1(x_1) ) = 0 11.97/4.25 POL( a__isNatKind_1(x_1) ) = 0 11.97/4.25 POL( U13_3(x_1, ..., x_3) ) = x_1 11.97/4.25 POL( a__U13_3(x_1, ..., x_3) ) = x_1 11.97/4.25 POL( U14_3(x_1, ..., x_3) ) = x_1 11.97/4.25 POL( a__U14_3(x_1, ..., x_3) ) = x_1 11.97/4.25 POL( U15_2(x_1, x_2) ) = 2x_1 11.97/4.25 POL( a__U15_2(x_1, x_2) ) = 2x_1 11.97/4.25 POL( isNat_1(x_1) ) = 0 11.97/4.25 POL( a__isNat_1(x_1) ) = 0 11.97/4.25 POL( U16_1(x_1) ) = 2x_1 11.97/4.25 POL( a__U16_1(x_1) ) = 2x_1 11.97/4.25 POL( U21_2(x_1, x_2) ) = x_1 11.97/4.25 POL( a__U21_2(x_1, x_2) ) = x_1 11.97/4.25 POL( U22_2(x_1, x_2) ) = 2x_1 11.97/4.25 POL( a__U22_2(x_1, x_2) ) = 2x_1 11.97/4.25 POL( U23_1(x_1) ) = x_1 11.97/4.25 POL( a__U23_1(x_1) ) = x_1 11.97/4.25 POL( U31_2(x_1, x_2) ) = x_1 11.97/4.25 POL( a__U31_2(x_1, x_2) ) = x_1 11.97/4.25 POL( U32_1(x_1) ) = x_1 11.97/4.25 POL( a__U32_1(x_1) ) = x_1 11.97/4.25 POL( U41_1(x_1) ) = x_1 11.97/4.25 POL( a__U41_1(x_1) ) = x_1 11.97/4.25 POL( U51_2(x_1, x_2) ) = x_1 + x_2 + 1 11.97/4.25 POL( a__U51_2(x_1, x_2) ) = x_1 + x_2 + 1 11.97/4.25 POL( tt ) = 0 11.97/4.25 POL( a__U52_2(x_1, x_2) ) = x_1 + x_2 + 1 11.97/4.25 POL( U52_2(x_1, x_2) ) = x_1 + x_2 + 1 11.97/4.25 POL( plus_2(x_1, x_2) ) = x_1 + 2x_2 11.97/4.25 POL( a__plus_2(x_1, x_2) ) = x_1 + 2x_2 11.97/4.25 POL( 0 ) = 1 11.97/4.25 POL( U61_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( a__U61_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( U62_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( a__U62_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( U63_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( a__U63_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( U64_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( a__U64_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 1 11.97/4.25 POL( s_1(x_1) ) = x_1 + 1 11.97/4.25 POL( MARK_1(x_1) ) = 2x_1 11.97/4.25 11.97/4.25 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 11.97/4.25 11.97/4.25 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.25 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.25 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.25 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.25 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.25 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.25 mark(isNat(X)) -> a__isNat(X) 11.97/4.25 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.25 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.25 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.25 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.25 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.25 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.25 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.25 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.25 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.25 a__U52(tt, N) -> mark(N) 11.97/4.25 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.25 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.25 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.25 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.25 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.25 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.25 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.25 mark(tt) -> tt 11.97/4.25 mark(s(X)) -> s(mark(X)) 11.97/4.25 mark(0) -> 0 11.97/4.25 a__isNatKind(0) -> tt 11.97/4.25 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.25 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.25 a__isNatKind(X) -> isNatKind(X) 11.97/4.25 a__isNat(0) -> tt 11.97/4.25 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.25 a__isNat(X) -> isNat(X) 11.97/4.25 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.25 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.25 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.25 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.25 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.25 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.25 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.25 a__U16(tt) -> tt 11.97/4.25 a__U16(X) -> U16(X) 11.97/4.25 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.25 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.25 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.25 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.25 a__U23(tt) -> tt 11.97/4.25 a__U23(X) -> U23(X) 11.97/4.25 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.25 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.25 a__U32(tt) -> tt 11.97/4.25 a__U32(X) -> U32(X) 11.97/4.25 a__U41(tt) -> tt 11.97/4.25 a__U41(X) -> U41(X) 11.97/4.25 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.25 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.25 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.25 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.25 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.25 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.25 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.25 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.25 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.25 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.25 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.25 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.25 11.97/4.25 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (21) 11.97/4.25 Obligation: 11.97/4.25 Q DP problem: 11.97/4.25 The TRS P consists of the following rules: 11.97/4.25 11.97/4.25 MARK(U11(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U12(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U13(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U14(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U15(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U16(X)) -> MARK(X) 11.97/4.25 MARK(U21(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U22(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U23(X)) -> MARK(X) 11.97/4.25 MARK(U31(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U32(X)) -> MARK(X) 11.97/4.25 MARK(U41(X)) -> MARK(X) 11.97/4.25 A__U51(tt, N) -> A__U52(a__isNatKind(N), N) 11.97/4.25 A__U61(tt, M, N) -> A__U62(a__isNatKind(M), M, N) 11.97/4.25 A__U62(tt, M, N) -> A__U63(a__isNat(N), M, N) 11.97/4.25 A__U64(tt, M, N) -> A__PLUS(mark(N), mark(M)) 11.97/4.25 A__U64(tt, M, N) -> MARK(N) 11.97/4.25 A__U64(tt, M, N) -> MARK(M) 11.97/4.25 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X2) 11.97/4.25 11.97/4.25 The TRS R consists of the following rules: 11.97/4.25 11.97/4.25 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.25 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.25 a__U16(tt) -> tt 11.97/4.25 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.25 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.25 a__U23(tt) -> tt 11.97/4.25 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.25 a__U32(tt) -> tt 11.97/4.25 a__U41(tt) -> tt 11.97/4.25 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.25 a__U52(tt, N) -> mark(N) 11.97/4.25 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.25 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.25 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.25 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.25 a__isNat(0) -> tt 11.97/4.25 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.25 a__isNatKind(0) -> tt 11.97/4.25 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.25 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.25 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.25 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.25 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.25 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.25 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.25 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.25 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.25 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.25 mark(isNat(X)) -> a__isNat(X) 11.97/4.25 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.25 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.25 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.25 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.25 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.25 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.25 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.25 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.25 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.25 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.25 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.25 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.25 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.25 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.25 mark(tt) -> tt 11.97/4.25 mark(s(X)) -> s(mark(X)) 11.97/4.25 mark(0) -> 0 11.97/4.25 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.25 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.25 a__isNatKind(X) -> isNatKind(X) 11.97/4.25 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.25 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.25 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.25 a__isNat(X) -> isNat(X) 11.97/4.25 a__U16(X) -> U16(X) 11.97/4.25 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.25 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.25 a__U23(X) -> U23(X) 11.97/4.25 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.25 a__U32(X) -> U32(X) 11.97/4.25 a__U41(X) -> U41(X) 11.97/4.25 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.25 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.25 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.25 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.25 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.25 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.25 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.25 11.97/4.25 The set Q consists of the following terms: 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (22) DependencyGraphProof (EQUIVALENT) 11.97/4.25 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (23) 11.97/4.25 Obligation: 11.97/4.25 Q DP problem: 11.97/4.25 The TRS P consists of the following rules: 11.97/4.25 11.97/4.25 MARK(U12(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U11(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U13(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U14(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U15(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U16(X)) -> MARK(X) 11.97/4.25 MARK(U21(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U22(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U23(X)) -> MARK(X) 11.97/4.25 MARK(U31(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U32(X)) -> MARK(X) 11.97/4.25 MARK(U41(X)) -> MARK(X) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X2) 11.97/4.25 11.97/4.25 The TRS R consists of the following rules: 11.97/4.25 11.97/4.25 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 11.97/4.25 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 11.97/4.25 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 11.97/4.25 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 11.97/4.25 a__U16(tt) -> tt 11.97/4.25 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 11.97/4.25 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 11.97/4.25 a__U23(tt) -> tt 11.97/4.25 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 11.97/4.25 a__U32(tt) -> tt 11.97/4.25 a__U41(tt) -> tt 11.97/4.25 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 11.97/4.25 a__U52(tt, N) -> mark(N) 11.97/4.25 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 11.97/4.25 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 11.97/4.25 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 11.97/4.25 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 11.97/4.25 a__isNat(0) -> tt 11.97/4.25 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 11.97/4.25 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 11.97/4.25 a__isNatKind(0) -> tt 11.97/4.25 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 11.97/4.25 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 11.97/4.25 a__plus(N, 0) -> a__U51(a__isNat(N), N) 11.97/4.25 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 11.97/4.25 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 11.97/4.25 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 11.97/4.25 mark(isNatKind(X)) -> a__isNatKind(X) 11.97/4.25 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 11.97/4.25 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 11.97/4.25 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 11.97/4.25 mark(isNat(X)) -> a__isNat(X) 11.97/4.25 mark(U16(X)) -> a__U16(mark(X)) 11.97/4.25 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 11.97/4.25 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 11.97/4.25 mark(U23(X)) -> a__U23(mark(X)) 11.97/4.25 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 11.97/4.25 mark(U32(X)) -> a__U32(mark(X)) 11.97/4.25 mark(U41(X)) -> a__U41(mark(X)) 11.97/4.25 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 11.97/4.25 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 11.97/4.25 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 11.97/4.25 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 11.97/4.25 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 11.97/4.25 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 11.97/4.25 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 11.97/4.25 mark(tt) -> tt 11.97/4.25 mark(s(X)) -> s(mark(X)) 11.97/4.25 mark(0) -> 0 11.97/4.25 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 11.97/4.25 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 11.97/4.25 a__isNatKind(X) -> isNatKind(X) 11.97/4.25 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 11.97/4.25 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 11.97/4.25 a__U15(X1, X2) -> U15(X1, X2) 11.97/4.25 a__isNat(X) -> isNat(X) 11.97/4.25 a__U16(X) -> U16(X) 11.97/4.25 a__U21(X1, X2) -> U21(X1, X2) 11.97/4.25 a__U22(X1, X2) -> U22(X1, X2) 11.97/4.25 a__U23(X) -> U23(X) 11.97/4.25 a__U31(X1, X2) -> U31(X1, X2) 11.97/4.25 a__U32(X) -> U32(X) 11.97/4.25 a__U41(X) -> U41(X) 11.97/4.25 a__U51(X1, X2) -> U51(X1, X2) 11.97/4.25 a__U52(X1, X2) -> U52(X1, X2) 11.97/4.25 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 11.97/4.25 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 11.97/4.25 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 11.97/4.25 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 11.97/4.25 a__plus(X1, X2) -> plus(X1, X2) 11.97/4.25 11.97/4.25 The set Q consists of the following terms: 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (24) UsableRulesProof (EQUIVALENT) 11.97/4.25 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. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (25) 11.97/4.25 Obligation: 11.97/4.25 Q DP problem: 11.97/4.25 The TRS P consists of the following rules: 11.97/4.25 11.97/4.25 MARK(U12(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U11(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U13(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U14(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U15(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U16(X)) -> MARK(X) 11.97/4.25 MARK(U21(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U22(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U23(X)) -> MARK(X) 11.97/4.25 MARK(U31(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U32(X)) -> MARK(X) 11.97/4.25 MARK(U41(X)) -> MARK(X) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X2) 11.97/4.25 11.97/4.25 R is empty. 11.97/4.25 The set Q consists of the following terms: 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (26) QReductionProof (EQUIVALENT) 11.97/4.25 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 11.97/4.25 11.97/4.25 mark(U11(x0, x1, x2)) 11.97/4.25 mark(U12(x0, x1, x2)) 11.97/4.25 mark(isNatKind(x0)) 11.97/4.25 mark(U13(x0, x1, x2)) 11.97/4.25 mark(U14(x0, x1, x2)) 11.97/4.25 mark(U15(x0, x1)) 11.97/4.25 mark(isNat(x0)) 11.97/4.25 mark(U16(x0)) 11.97/4.25 mark(U21(x0, x1)) 11.97/4.25 mark(U22(x0, x1)) 11.97/4.25 mark(U23(x0)) 11.97/4.25 mark(U31(x0, x1)) 11.97/4.25 mark(U32(x0)) 11.97/4.25 mark(U41(x0)) 11.97/4.25 mark(U51(x0, x1)) 11.97/4.25 mark(U52(x0, x1)) 11.97/4.25 mark(U61(x0, x1, x2)) 11.97/4.25 mark(U62(x0, x1, x2)) 11.97/4.25 mark(U63(x0, x1, x2)) 11.97/4.25 mark(U64(x0, x1, x2)) 11.97/4.25 mark(plus(x0, x1)) 11.97/4.25 mark(tt) 11.97/4.25 mark(s(x0)) 11.97/4.25 mark(0) 11.97/4.25 a__U11(x0, x1, x2) 11.97/4.25 a__U12(x0, x1, x2) 11.97/4.25 a__isNatKind(x0) 11.97/4.25 a__U13(x0, x1, x2) 11.97/4.25 a__U14(x0, x1, x2) 11.97/4.25 a__U15(x0, x1) 11.97/4.25 a__isNat(x0) 11.97/4.25 a__U16(x0) 11.97/4.25 a__U21(x0, x1) 11.97/4.25 a__U22(x0, x1) 11.97/4.25 a__U23(x0) 11.97/4.25 a__U31(x0, x1) 11.97/4.25 a__U32(x0) 11.97/4.25 a__U41(x0) 11.97/4.25 a__U51(x0, x1) 11.97/4.25 a__U52(x0, x1) 11.97/4.25 a__U61(x0, x1, x2) 11.97/4.25 a__U62(x0, x1, x2) 11.97/4.25 a__U63(x0, x1, x2) 11.97/4.25 a__U64(x0, x1, x2) 11.97/4.25 a__plus(x0, x1) 11.97/4.25 11.97/4.25 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (27) 11.97/4.25 Obligation: 11.97/4.25 Q DP problem: 11.97/4.25 The TRS P consists of the following rules: 11.97/4.25 11.97/4.25 MARK(U12(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U11(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U13(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U14(X1, X2, X3)) -> MARK(X1) 11.97/4.25 MARK(U15(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U16(X)) -> MARK(X) 11.97/4.25 MARK(U21(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U22(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U23(X)) -> MARK(X) 11.97/4.25 MARK(U31(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(U32(X)) -> MARK(X) 11.97/4.25 MARK(U41(X)) -> MARK(X) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X1) 11.97/4.25 MARK(plus(X1, X2)) -> MARK(X2) 11.97/4.25 11.97/4.25 R is empty. 11.97/4.25 Q is empty. 11.97/4.25 We have to consider all minimal (P,Q,R)-chains. 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (28) QDPSizeChangeProof (EQUIVALENT) 11.97/4.25 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. 11.97/4.25 11.97/4.25 From the DPs we obtained the following set of size-change graphs: 11.97/4.25 *MARK(U12(X1, X2, X3)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U11(X1, X2, X3)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U13(X1, X2, X3)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U14(X1, X2, X3)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U15(X1, X2)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U16(X)) -> MARK(X) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U21(X1, X2)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U22(X1, X2)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U23(X)) -> MARK(X) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U31(X1, X2)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U32(X)) -> MARK(X) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(U41(X)) -> MARK(X) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(plus(X1, X2)) -> MARK(X1) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 *MARK(plus(X1, X2)) -> MARK(X2) 11.97/4.25 The graph contains the following edges 1 > 1 11.97/4.25 11.97/4.25 11.97/4.25 ---------------------------------------- 11.97/4.25 11.97/4.25 (29) 11.97/4.25 YES 12.29/4.49 EOF