42.11/12.83 YES 42.39/12.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 42.39/12.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 42.39/12.85 42.39/12.85 42.39/12.85 Termination w.r.t. Q of the given QTRS could be proven: 42.39/12.85 42.39/12.85 (0) QTRS 42.39/12.85 (1) DependencyPairsProof [EQUIVALENT, 239 ms] 42.39/12.85 (2) QDP 42.39/12.85 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 42.39/12.85 (4) AND 42.39/12.85 (5) QDP 42.39/12.85 (6) UsableRulesProof [EQUIVALENT, 0 ms] 42.39/12.85 (7) QDP 42.39/12.85 (8) QReductionProof [EQUIVALENT, 0 ms] 42.39/12.85 (9) QDP 42.39/12.85 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.39/12.85 (11) YES 42.39/12.85 (12) QDP 42.39/12.85 (13) UsableRulesProof [EQUIVALENT, 0 ms] 42.39/12.85 (14) QDP 42.39/12.85 (15) QReductionProof [EQUIVALENT, 0 ms] 42.39/12.85 (16) QDP 42.39/12.85 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.39/12.85 (18) YES 42.39/12.85 (19) QDP 42.39/12.85 (20) QDPOrderProof [EQUIVALENT, 6374 ms] 42.39/12.85 (21) QDP 42.39/12.85 (22) DependencyGraphProof [EQUIVALENT, 0 ms] 42.39/12.85 (23) QDP 42.39/12.85 (24) UsableRulesProof [EQUIVALENT, 0 ms] 42.39/12.85 (25) QDP 42.39/12.85 (26) QReductionProof [EQUIVALENT, 0 ms] 42.39/12.85 (27) QDP 42.39/12.85 (28) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.39/12.85 (29) YES 42.39/12.85 42.39/12.85 42.39/12.85 ---------------------------------------- 42.39/12.85 42.39/12.85 (0) 42.39/12.85 Obligation: 42.39/12.85 Q restricted rewrite system: 42.39/12.85 The TRS R consists of the following rules: 42.39/12.85 42.39/12.85 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.85 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.85 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.85 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.85 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.85 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.85 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.85 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.85 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.85 a__U16(tt) -> tt 42.39/12.85 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.85 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.85 a__U23(tt) -> tt 42.39/12.85 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.85 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.85 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.85 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.85 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.85 a__U36(tt) -> tt 42.39/12.85 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.85 a__U42(tt) -> tt 42.39/12.85 a__U51(tt) -> tt 42.39/12.85 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.85 a__U62(tt) -> tt 42.39/12.85 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.85 a__U72(tt, N) -> mark(N) 42.39/12.85 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.85 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.85 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.85 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.85 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.85 a__U92(tt) -> 0 42.39/12.85 a__isNat(0) -> tt 42.39/12.85 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.85 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.85 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.85 a__isNatKind(0) -> tt 42.39/12.85 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.85 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.85 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.85 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.85 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.85 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.85 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.85 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.85 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.85 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.85 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.85 mark(isNat(X)) -> a__isNat(X) 42.39/12.85 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.85 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.85 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.85 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.85 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.85 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.85 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.85 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.85 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.85 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.85 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.85 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.85 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.85 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.85 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.85 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.85 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.85 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.85 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.85 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.85 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.85 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.85 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.85 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.85 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.85 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.85 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.85 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.85 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.85 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.85 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.85 mark(tt) -> tt 42.39/12.85 mark(s(X)) -> s(mark(X)) 42.39/12.85 mark(0) -> 0 42.39/12.85 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.85 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.85 a__isNatKind(X) -> isNatKind(X) 42.39/12.85 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.85 a__isNat(X) -> isNat(X) 42.39/12.85 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.85 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.85 a__x(X1, X2) -> x(X1, X2) 42.39/12.85 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.85 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.85 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.85 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.85 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.85 a__U16(X) -> U16(X) 42.39/12.85 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.85 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.85 a__U23(X) -> U23(X) 42.39/12.85 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.85 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.85 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.85 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.85 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.85 a__U36(X) -> U36(X) 42.39/12.85 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.85 a__U42(X) -> U42(X) 42.39/12.85 a__U51(X) -> U51(X) 42.39/12.85 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.85 a__U62(X) -> U62(X) 42.39/12.85 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.85 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.85 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.85 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.85 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.85 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.85 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.85 a__U92(X) -> U92(X) 42.39/12.85 42.39/12.85 The set Q consists of the following terms: 42.39/12.85 42.39/12.85 mark(U101(x0, x1, x2)) 42.39/12.85 mark(U102(x0, x1, x2)) 42.39/12.85 mark(isNatKind(x0)) 42.39/12.85 mark(U103(x0, x1, x2)) 42.39/12.85 mark(isNat(x0)) 42.39/12.85 mark(U104(x0, x1, x2)) 42.39/12.85 mark(plus(x0, x1)) 42.39/12.85 mark(x(x0, x1)) 42.39/12.85 mark(U11(x0, x1, x2)) 42.39/12.85 mark(U12(x0, x1, x2)) 42.39/12.85 mark(U13(x0, x1, x2)) 42.39/12.85 mark(U14(x0, x1, x2)) 42.39/12.85 mark(U15(x0, x1)) 42.39/12.85 mark(U16(x0)) 42.39/12.85 mark(U21(x0, x1)) 42.39/12.85 mark(U22(x0, x1)) 42.39/12.85 mark(U23(x0)) 42.39/12.85 mark(U31(x0, x1, x2)) 42.39/12.85 mark(U32(x0, x1, x2)) 42.39/12.85 mark(U33(x0, x1, x2)) 42.39/12.85 mark(U34(x0, x1, x2)) 42.39/12.85 mark(U35(x0, x1)) 42.39/12.85 mark(U36(x0)) 42.39/12.85 mark(U41(x0, x1)) 42.39/12.85 mark(U42(x0)) 42.39/12.85 mark(U51(x0)) 42.39/12.85 mark(U61(x0, x1)) 42.39/12.85 mark(U62(x0)) 42.39/12.85 mark(U71(x0, x1)) 42.39/12.85 mark(U72(x0, x1)) 42.39/12.85 mark(U81(x0, x1, x2)) 42.39/12.85 mark(U82(x0, x1, x2)) 42.39/12.85 mark(U83(x0, x1, x2)) 42.39/12.85 mark(U84(x0, x1, x2)) 42.39/12.85 mark(U91(x0, x1)) 42.39/12.85 mark(U92(x0)) 42.39/12.85 mark(tt) 42.39/12.85 mark(s(x0)) 42.39/12.85 mark(0) 42.39/12.85 a__U101(x0, x1, x2) 42.39/12.85 a__U102(x0, x1, x2) 42.39/12.85 a__isNatKind(x0) 42.39/12.85 a__U103(x0, x1, x2) 42.39/12.85 a__isNat(x0) 42.39/12.85 a__U104(x0, x1, x2) 42.39/12.85 a__plus(x0, x1) 42.39/12.85 a__x(x0, x1) 42.39/12.85 a__U11(x0, x1, x2) 42.39/12.85 a__U12(x0, x1, x2) 42.39/12.85 a__U13(x0, x1, x2) 42.39/12.85 a__U14(x0, x1, x2) 42.39/12.85 a__U15(x0, x1) 42.39/12.85 a__U16(x0) 42.39/12.85 a__U21(x0, x1) 42.39/12.85 a__U22(x0, x1) 42.39/12.85 a__U23(x0) 42.39/12.85 a__U31(x0, x1, x2) 42.39/12.85 a__U32(x0, x1, x2) 42.39/12.85 a__U33(x0, x1, x2) 42.39/12.85 a__U34(x0, x1, x2) 42.39/12.85 a__U35(x0, x1) 42.39/12.85 a__U36(x0) 42.39/12.85 a__U41(x0, x1) 42.39/12.85 a__U42(x0) 42.39/12.85 a__U51(x0) 42.39/12.85 a__U61(x0, x1) 42.39/12.85 a__U62(x0) 42.39/12.85 a__U71(x0, x1) 42.39/12.85 a__U72(x0, x1) 42.39/12.85 a__U81(x0, x1, x2) 42.39/12.85 a__U82(x0, x1, x2) 42.39/12.85 a__U83(x0, x1, x2) 42.39/12.85 a__U84(x0, x1, x2) 42.39/12.85 a__U91(x0, x1) 42.39/12.85 a__U92(x0) 42.39/12.85 42.39/12.85 42.39/12.85 ---------------------------------------- 42.39/12.85 42.39/12.85 (1) DependencyPairsProof (EQUIVALENT) 42.39/12.85 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 42.39/12.85 ---------------------------------------- 42.39/12.85 42.39/12.85 (2) 42.39/12.85 Obligation: 42.39/12.85 Q DP problem: 42.39/12.85 The TRS P consists of the following rules: 42.39/12.85 42.39/12.85 A__U101(tt, M, N) -> A__U102(a__isNatKind(M), M, N) 42.39/12.85 A__U101(tt, M, N) -> A__ISNATKIND(M) 42.39/12.85 A__U102(tt, M, N) -> A__U103(a__isNat(N), M, N) 42.39/12.85 A__U102(tt, M, N) -> A__ISNAT(N) 42.39/12.85 A__U103(tt, M, N) -> A__U104(a__isNatKind(N), M, N) 42.39/12.85 A__U103(tt, M, N) -> A__ISNATKIND(N) 42.39/12.85 A__U104(tt, M, N) -> A__PLUS(a__x(mark(N), mark(M)), mark(N)) 42.39/12.85 A__U104(tt, M, N) -> A__X(mark(N), mark(M)) 42.39/12.85 A__U104(tt, M, N) -> MARK(N) 42.39/12.85 A__U104(tt, M, N) -> MARK(M) 42.39/12.85 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 42.39/12.85 A__U11(tt, V1, V2) -> A__ISNATKIND(V1) 42.39/12.85 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 42.39/12.85 A__U12(tt, V1, V2) -> A__ISNATKIND(V2) 42.39/12.85 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 42.39/12.85 A__U13(tt, V1, V2) -> A__ISNATKIND(V2) 42.39/12.85 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 42.39/12.85 A__U14(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.85 A__U15(tt, V2) -> A__U16(a__isNat(V2)) 42.39/12.85 A__U15(tt, V2) -> A__ISNAT(V2) 42.39/12.85 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.39/12.85 A__U21(tt, V1) -> A__ISNATKIND(V1) 42.39/12.85 A__U22(tt, V1) -> A__U23(a__isNat(V1)) 42.39/12.85 A__U22(tt, V1) -> A__ISNAT(V1) 42.39/12.85 A__U31(tt, V1, V2) -> A__U32(a__isNatKind(V1), V1, V2) 42.39/12.85 A__U31(tt, V1, V2) -> A__ISNATKIND(V1) 42.39/12.85 A__U32(tt, V1, V2) -> A__U33(a__isNatKind(V2), V1, V2) 42.39/12.85 A__U32(tt, V1, V2) -> A__ISNATKIND(V2) 42.39/12.85 A__U33(tt, V1, V2) -> A__U34(a__isNatKind(V2), V1, V2) 42.39/12.85 A__U33(tt, V1, V2) -> A__ISNATKIND(V2) 42.39/12.85 A__U34(tt, V1, V2) -> A__U35(a__isNat(V1), V2) 42.39/12.85 A__U34(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.85 A__U35(tt, V2) -> A__U36(a__isNat(V2)) 42.39/12.85 A__U35(tt, V2) -> A__ISNAT(V2) 42.39/12.85 A__U41(tt, V2) -> A__U42(a__isNatKind(V2)) 42.39/12.85 A__U41(tt, V2) -> A__ISNATKIND(V2) 42.39/12.85 A__U61(tt, V2) -> A__U62(a__isNatKind(V2)) 42.39/12.85 A__U61(tt, V2) -> A__ISNATKIND(V2) 42.39/12.85 A__U71(tt, N) -> A__U72(a__isNatKind(N), N) 42.39/12.85 A__U71(tt, N) -> A__ISNATKIND(N) 42.39/12.85 A__U72(tt, N) -> MARK(N) 42.39/12.85 A__U81(tt, M, N) -> A__U82(a__isNatKind(M), M, N) 42.39/12.85 A__U81(tt, M, N) -> A__ISNATKIND(M) 42.39/12.85 A__U82(tt, M, N) -> A__U83(a__isNat(N), M, N) 42.39/12.85 A__U82(tt, M, N) -> A__ISNAT(N) 42.39/12.85 A__U83(tt, M, N) -> A__U84(a__isNatKind(N), M, N) 42.39/12.85 A__U83(tt, M, N) -> A__ISNATKIND(N) 42.39/12.85 A__U84(tt, M, N) -> A__PLUS(mark(N), mark(M)) 42.39/12.85 A__U84(tt, M, N) -> MARK(N) 42.39/12.85 A__U84(tt, M, N) -> MARK(M) 42.39/12.85 A__U91(tt, N) -> A__U92(a__isNatKind(N)) 42.39/12.85 A__U91(tt, N) -> A__ISNATKIND(N) 42.39/12.85 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 42.39/12.85 A__ISNAT(plus(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.85 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.39/12.85 A__ISNAT(s(V1)) -> A__ISNATKIND(V1) 42.39/12.85 A__ISNAT(x(V1, V2)) -> A__U31(a__isNatKind(V1), V1, V2) 42.39/12.85 A__ISNAT(x(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.85 A__ISNATKIND(plus(V1, V2)) -> A__U41(a__isNatKind(V1), V2) 42.39/12.85 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.85 A__ISNATKIND(s(V1)) -> A__U51(a__isNatKind(V1)) 42.39/12.85 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.39/12.85 A__ISNATKIND(x(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.39/12.85 A__ISNATKIND(x(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.85 A__PLUS(N, 0) -> A__U71(a__isNat(N), N) 42.39/12.85 A__PLUS(N, 0) -> A__ISNAT(N) 42.39/12.85 A__PLUS(N, s(M)) -> A__U81(a__isNat(M), M, N) 42.39/12.85 A__PLUS(N, s(M)) -> A__ISNAT(M) 42.39/12.85 A__X(N, 0) -> A__U91(a__isNat(N), N) 42.39/12.85 A__X(N, 0) -> A__ISNAT(N) 42.39/12.85 A__X(N, s(M)) -> A__U101(a__isNat(M), M, N) 42.39/12.85 A__X(N, s(M)) -> A__ISNAT(M) 42.39/12.85 MARK(U101(X1, X2, X3)) -> A__U101(mark(X1), X2, X3) 42.39/12.86 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U102(X1, X2, X3)) -> A__U102(mark(X1), X2, X3) 42.39/12.86 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(isNatKind(X)) -> A__ISNATKIND(X) 42.39/12.86 MARK(U103(X1, X2, X3)) -> A__U103(mark(X1), X2, X3) 42.39/12.86 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(isNat(X)) -> A__ISNAT(X) 42.39/12.86 MARK(U104(X1, X2, X3)) -> A__U104(mark(X1), X2, X3) 42.39/12.86 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 42.39/12.86 MARK(plus(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(plus(X1, X2)) -> MARK(X2) 42.39/12.86 MARK(x(X1, X2)) -> A__X(mark(X1), mark(X2)) 42.39/12.86 MARK(x(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(x(X1, X2)) -> MARK(X2) 42.39/12.86 MARK(U11(X1, X2, X3)) -> A__U11(mark(X1), X2, X3) 42.39/12.86 MARK(U11(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U12(X1, X2, X3)) -> A__U12(mark(X1), X2, X3) 42.39/12.86 MARK(U12(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U13(X1, X2, X3)) -> A__U13(mark(X1), X2, X3) 42.39/12.86 MARK(U13(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U14(X1, X2, X3)) -> A__U14(mark(X1), X2, X3) 42.39/12.86 MARK(U14(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U15(X1, X2)) -> A__U15(mark(X1), X2) 42.39/12.86 MARK(U15(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U16(X)) -> A__U16(mark(X)) 42.39/12.86 MARK(U16(X)) -> MARK(X) 42.39/12.86 MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) 42.39/12.86 MARK(U21(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U22(X1, X2)) -> A__U22(mark(X1), X2) 42.39/12.86 MARK(U22(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U23(X)) -> A__U23(mark(X)) 42.39/12.86 MARK(U23(X)) -> MARK(X) 42.39/12.86 MARK(U31(X1, X2, X3)) -> A__U31(mark(X1), X2, X3) 42.39/12.86 MARK(U31(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U32(X1, X2, X3)) -> A__U32(mark(X1), X2, X3) 42.39/12.86 MARK(U32(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U33(X1, X2, X3)) -> A__U33(mark(X1), X2, X3) 42.39/12.86 MARK(U33(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U34(X1, X2, X3)) -> A__U34(mark(X1), X2, X3) 42.39/12.86 MARK(U34(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U35(X1, X2)) -> A__U35(mark(X1), X2) 42.39/12.86 MARK(U35(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U36(X)) -> A__U36(mark(X)) 42.39/12.86 MARK(U36(X)) -> MARK(X) 42.39/12.86 MARK(U41(X1, X2)) -> A__U41(mark(X1), X2) 42.39/12.86 MARK(U41(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U42(X)) -> A__U42(mark(X)) 42.39/12.86 MARK(U42(X)) -> MARK(X) 42.39/12.86 MARK(U51(X)) -> A__U51(mark(X)) 42.39/12.86 MARK(U51(X)) -> MARK(X) 42.39/12.86 MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) 42.39/12.86 MARK(U61(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U62(X)) -> A__U62(mark(X)) 42.39/12.86 MARK(U62(X)) -> MARK(X) 42.39/12.86 MARK(U71(X1, X2)) -> A__U71(mark(X1), X2) 42.39/12.86 MARK(U71(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U72(X1, X2)) -> A__U72(mark(X1), X2) 42.39/12.86 MARK(U72(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U81(X1, X2, X3)) -> A__U81(mark(X1), X2, X3) 42.39/12.86 MARK(U81(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U82(X1, X2, X3)) -> A__U82(mark(X1), X2, X3) 42.39/12.86 MARK(U82(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U83(X1, X2, X3)) -> A__U83(mark(X1), X2, X3) 42.39/12.86 MARK(U83(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U84(X1, X2, X3)) -> A__U84(mark(X1), X2, X3) 42.39/12.86 MARK(U84(X1, X2, X3)) -> MARK(X1) 42.39/12.86 MARK(U91(X1, X2)) -> A__U91(mark(X1), X2) 42.39/12.86 MARK(U91(X1, X2)) -> MARK(X1) 42.39/12.86 MARK(U92(X)) -> A__U92(mark(X)) 42.39/12.86 MARK(U92(X)) -> MARK(X) 42.39/12.86 MARK(s(X)) -> MARK(X) 42.39/12.86 42.39/12.86 The TRS R consists of the following rules: 42.39/12.86 42.39/12.86 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.86 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.86 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.86 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.86 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.86 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.86 a__U16(tt) -> tt 42.39/12.86 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.86 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.86 a__U23(tt) -> tt 42.39/12.86 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.86 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.86 a__U36(tt) -> tt 42.39/12.86 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.86 a__U42(tt) -> tt 42.39/12.86 a__U51(tt) -> tt 42.39/12.86 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.86 a__U62(tt) -> tt 42.39/12.86 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.86 a__U72(tt, N) -> mark(N) 42.39/12.86 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.86 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.86 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.86 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.86 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.86 a__U92(tt) -> 0 42.39/12.86 a__isNat(0) -> tt 42.39/12.86 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.86 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNatKind(0) -> tt 42.39/12.86 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.86 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.86 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.86 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.86 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.86 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.86 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.86 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.86 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.86 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.86 mark(isNat(X)) -> a__isNat(X) 42.39/12.86 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.86 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.86 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.86 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.86 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.86 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.86 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.86 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.86 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.86 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.86 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.86 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.86 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.86 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.86 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.86 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.86 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.86 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.86 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.86 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.86 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.86 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.86 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.86 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.86 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.86 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.86 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.86 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.86 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.86 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.86 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.86 mark(tt) -> tt 42.39/12.86 mark(s(X)) -> s(mark(X)) 42.39/12.86 mark(0) -> 0 42.39/12.86 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.86 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.86 a__isNatKind(X) -> isNatKind(X) 42.39/12.86 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.86 a__isNat(X) -> isNat(X) 42.39/12.86 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.86 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.86 a__x(X1, X2) -> x(X1, X2) 42.39/12.86 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.86 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.86 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.86 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.86 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.86 a__U16(X) -> U16(X) 42.39/12.86 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.86 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.86 a__U23(X) -> U23(X) 42.39/12.86 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.86 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.86 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.86 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.86 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.86 a__U36(X) -> U36(X) 42.39/12.86 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.86 a__U42(X) -> U42(X) 42.39/12.86 a__U51(X) -> U51(X) 42.39/12.86 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.86 a__U62(X) -> U62(X) 42.39/12.86 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.86 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.86 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.86 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.86 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.86 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.86 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.86 a__U92(X) -> U92(X) 42.39/12.86 42.39/12.86 The set Q consists of the following terms: 42.39/12.86 42.39/12.86 mark(U101(x0, x1, x2)) 42.39/12.86 mark(U102(x0, x1, x2)) 42.39/12.86 mark(isNatKind(x0)) 42.39/12.86 mark(U103(x0, x1, x2)) 42.39/12.86 mark(isNat(x0)) 42.39/12.86 mark(U104(x0, x1, x2)) 42.39/12.86 mark(plus(x0, x1)) 42.39/12.86 mark(x(x0, x1)) 42.39/12.86 mark(U11(x0, x1, x2)) 42.39/12.86 mark(U12(x0, x1, x2)) 42.39/12.86 mark(U13(x0, x1, x2)) 42.39/12.86 mark(U14(x0, x1, x2)) 42.39/12.86 mark(U15(x0, x1)) 42.39/12.86 mark(U16(x0)) 42.39/12.86 mark(U21(x0, x1)) 42.39/12.86 mark(U22(x0, x1)) 42.39/12.86 mark(U23(x0)) 42.39/12.86 mark(U31(x0, x1, x2)) 42.39/12.86 mark(U32(x0, x1, x2)) 42.39/12.86 mark(U33(x0, x1, x2)) 42.39/12.86 mark(U34(x0, x1, x2)) 42.39/12.86 mark(U35(x0, x1)) 42.39/12.86 mark(U36(x0)) 42.39/12.86 mark(U41(x0, x1)) 42.39/12.86 mark(U42(x0)) 42.39/12.86 mark(U51(x0)) 42.39/12.86 mark(U61(x0, x1)) 42.39/12.86 mark(U62(x0)) 42.39/12.86 mark(U71(x0, x1)) 42.39/12.86 mark(U72(x0, x1)) 42.39/12.86 mark(U81(x0, x1, x2)) 42.39/12.86 mark(U82(x0, x1, x2)) 42.39/12.86 mark(U83(x0, x1, x2)) 42.39/12.86 mark(U84(x0, x1, x2)) 42.39/12.86 mark(U91(x0, x1)) 42.39/12.86 mark(U92(x0)) 42.39/12.86 mark(tt) 42.39/12.86 mark(s(x0)) 42.39/12.86 mark(0) 42.39/12.86 a__U101(x0, x1, x2) 42.39/12.86 a__U102(x0, x1, x2) 42.39/12.86 a__isNatKind(x0) 42.39/12.86 a__U103(x0, x1, x2) 42.39/12.86 a__isNat(x0) 42.39/12.86 a__U104(x0, x1, x2) 42.39/12.86 a__plus(x0, x1) 42.39/12.86 a__x(x0, x1) 42.39/12.86 a__U11(x0, x1, x2) 42.39/12.86 a__U12(x0, x1, x2) 42.39/12.86 a__U13(x0, x1, x2) 42.39/12.86 a__U14(x0, x1, x2) 42.39/12.86 a__U15(x0, x1) 42.39/12.86 a__U16(x0) 42.39/12.86 a__U21(x0, x1) 42.39/12.86 a__U22(x0, x1) 42.39/12.86 a__U23(x0) 42.39/12.86 a__U31(x0, x1, x2) 42.39/12.86 a__U32(x0, x1, x2) 42.39/12.86 a__U33(x0, x1, x2) 42.39/12.86 a__U34(x0, x1, x2) 42.39/12.86 a__U35(x0, x1) 42.39/12.86 a__U36(x0) 42.39/12.86 a__U41(x0, x1) 42.39/12.86 a__U42(x0) 42.39/12.86 a__U51(x0) 42.39/12.86 a__U61(x0, x1) 42.39/12.86 a__U62(x0) 42.39/12.86 a__U71(x0, x1) 42.39/12.86 a__U72(x0, x1) 42.39/12.86 a__U81(x0, x1, x2) 42.39/12.86 a__U82(x0, x1, x2) 42.39/12.86 a__U83(x0, x1, x2) 42.39/12.86 a__U84(x0, x1, x2) 42.39/12.86 a__U91(x0, x1) 42.39/12.86 a__U92(x0) 42.39/12.86 42.39/12.86 We have to consider all minimal (P,Q,R)-chains. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (3) DependencyGraphProof (EQUIVALENT) 42.39/12.86 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 54 less nodes. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (4) 42.39/12.86 Complex Obligation (AND) 42.39/12.86 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (5) 42.39/12.86 Obligation: 42.39/12.86 Q DP problem: 42.39/12.86 The TRS P consists of the following rules: 42.39/12.86 42.39/12.86 A__U41(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 A__ISNATKIND(plus(V1, V2)) -> A__U41(a__isNatKind(V1), V2) 42.39/12.86 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.39/12.86 A__ISNATKIND(x(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.39/12.86 A__U61(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 A__ISNATKIND(x(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 42.39/12.86 The TRS R consists of the following rules: 42.39/12.86 42.39/12.86 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.86 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.86 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.86 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.86 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.86 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.86 a__U16(tt) -> tt 42.39/12.86 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.86 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.86 a__U23(tt) -> tt 42.39/12.86 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.86 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.86 a__U36(tt) -> tt 42.39/12.86 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.86 a__U42(tt) -> tt 42.39/12.86 a__U51(tt) -> tt 42.39/12.86 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.86 a__U62(tt) -> tt 42.39/12.86 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.86 a__U72(tt, N) -> mark(N) 42.39/12.86 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.86 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.86 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.86 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.86 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.86 a__U92(tt) -> 0 42.39/12.86 a__isNat(0) -> tt 42.39/12.86 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.86 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNatKind(0) -> tt 42.39/12.86 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.86 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.86 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.86 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.86 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.86 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.86 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.86 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.86 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.86 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.86 mark(isNat(X)) -> a__isNat(X) 42.39/12.86 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.86 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.86 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.86 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.86 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.86 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.86 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.86 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.86 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.86 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.86 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.86 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.86 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.86 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.86 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.86 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.86 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.86 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.86 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.86 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.86 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.86 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.86 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.86 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.86 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.86 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.86 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.86 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.86 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.86 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.86 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.86 mark(tt) -> tt 42.39/12.86 mark(s(X)) -> s(mark(X)) 42.39/12.86 mark(0) -> 0 42.39/12.86 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.86 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.86 a__isNatKind(X) -> isNatKind(X) 42.39/12.86 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.86 a__isNat(X) -> isNat(X) 42.39/12.86 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.86 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.86 a__x(X1, X2) -> x(X1, X2) 42.39/12.86 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.86 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.86 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.86 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.86 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.86 a__U16(X) -> U16(X) 42.39/12.86 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.86 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.86 a__U23(X) -> U23(X) 42.39/12.86 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.86 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.86 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.86 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.86 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.86 a__U36(X) -> U36(X) 42.39/12.86 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.86 a__U42(X) -> U42(X) 42.39/12.86 a__U51(X) -> U51(X) 42.39/12.86 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.86 a__U62(X) -> U62(X) 42.39/12.86 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.86 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.86 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.86 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.86 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.86 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.86 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.86 a__U92(X) -> U92(X) 42.39/12.86 42.39/12.86 The set Q consists of the following terms: 42.39/12.86 42.39/12.86 mark(U101(x0, x1, x2)) 42.39/12.86 mark(U102(x0, x1, x2)) 42.39/12.86 mark(isNatKind(x0)) 42.39/12.86 mark(U103(x0, x1, x2)) 42.39/12.86 mark(isNat(x0)) 42.39/12.86 mark(U104(x0, x1, x2)) 42.39/12.86 mark(plus(x0, x1)) 42.39/12.86 mark(x(x0, x1)) 42.39/12.86 mark(U11(x0, x1, x2)) 42.39/12.86 mark(U12(x0, x1, x2)) 42.39/12.86 mark(U13(x0, x1, x2)) 42.39/12.86 mark(U14(x0, x1, x2)) 42.39/12.86 mark(U15(x0, x1)) 42.39/12.86 mark(U16(x0)) 42.39/12.86 mark(U21(x0, x1)) 42.39/12.86 mark(U22(x0, x1)) 42.39/12.86 mark(U23(x0)) 42.39/12.86 mark(U31(x0, x1, x2)) 42.39/12.86 mark(U32(x0, x1, x2)) 42.39/12.86 mark(U33(x0, x1, x2)) 42.39/12.86 mark(U34(x0, x1, x2)) 42.39/12.86 mark(U35(x0, x1)) 42.39/12.86 mark(U36(x0)) 42.39/12.86 mark(U41(x0, x1)) 42.39/12.86 mark(U42(x0)) 42.39/12.86 mark(U51(x0)) 42.39/12.86 mark(U61(x0, x1)) 42.39/12.86 mark(U62(x0)) 42.39/12.86 mark(U71(x0, x1)) 42.39/12.86 mark(U72(x0, x1)) 42.39/12.86 mark(U81(x0, x1, x2)) 42.39/12.86 mark(U82(x0, x1, x2)) 42.39/12.86 mark(U83(x0, x1, x2)) 42.39/12.86 mark(U84(x0, x1, x2)) 42.39/12.86 mark(U91(x0, x1)) 42.39/12.86 mark(U92(x0)) 42.39/12.86 mark(tt) 42.39/12.86 mark(s(x0)) 42.39/12.86 mark(0) 42.39/12.86 a__U101(x0, x1, x2) 42.39/12.86 a__U102(x0, x1, x2) 42.39/12.86 a__isNatKind(x0) 42.39/12.86 a__U103(x0, x1, x2) 42.39/12.86 a__isNat(x0) 42.39/12.86 a__U104(x0, x1, x2) 42.39/12.86 a__plus(x0, x1) 42.39/12.86 a__x(x0, x1) 42.39/12.86 a__U11(x0, x1, x2) 42.39/12.86 a__U12(x0, x1, x2) 42.39/12.86 a__U13(x0, x1, x2) 42.39/12.86 a__U14(x0, x1, x2) 42.39/12.86 a__U15(x0, x1) 42.39/12.86 a__U16(x0) 42.39/12.86 a__U21(x0, x1) 42.39/12.86 a__U22(x0, x1) 42.39/12.86 a__U23(x0) 42.39/12.86 a__U31(x0, x1, x2) 42.39/12.86 a__U32(x0, x1, x2) 42.39/12.86 a__U33(x0, x1, x2) 42.39/12.86 a__U34(x0, x1, x2) 42.39/12.86 a__U35(x0, x1) 42.39/12.86 a__U36(x0) 42.39/12.86 a__U41(x0, x1) 42.39/12.86 a__U42(x0) 42.39/12.86 a__U51(x0) 42.39/12.86 a__U61(x0, x1) 42.39/12.86 a__U62(x0) 42.39/12.86 a__U71(x0, x1) 42.39/12.86 a__U72(x0, x1) 42.39/12.86 a__U81(x0, x1, x2) 42.39/12.86 a__U82(x0, x1, x2) 42.39/12.86 a__U83(x0, x1, x2) 42.39/12.86 a__U84(x0, x1, x2) 42.39/12.86 a__U91(x0, x1) 42.39/12.86 a__U92(x0) 42.39/12.86 42.39/12.86 We have to consider all minimal (P,Q,R)-chains. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (6) UsableRulesProof (EQUIVALENT) 42.39/12.86 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. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (7) 42.39/12.86 Obligation: 42.39/12.86 Q DP problem: 42.39/12.86 The TRS P consists of the following rules: 42.39/12.86 42.39/12.86 A__U41(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 A__ISNATKIND(plus(V1, V2)) -> A__U41(a__isNatKind(V1), V2) 42.39/12.86 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.39/12.86 A__ISNATKIND(x(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.39/12.86 A__U61(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 A__ISNATKIND(x(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 42.39/12.86 The TRS R consists of the following rules: 42.39/12.86 42.39/12.86 a__isNatKind(0) -> tt 42.39/12.86 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.86 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(X) -> isNatKind(X) 42.39/12.86 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.86 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.86 a__U62(tt) -> tt 42.39/12.86 a__U62(X) -> U62(X) 42.39/12.86 a__U51(tt) -> tt 42.39/12.86 a__U51(X) -> U51(X) 42.39/12.86 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.86 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.86 a__U42(tt) -> tt 42.39/12.86 a__U42(X) -> U42(X) 42.39/12.86 42.39/12.86 The set Q consists of the following terms: 42.39/12.86 42.39/12.86 mark(U101(x0, x1, x2)) 42.39/12.86 mark(U102(x0, x1, x2)) 42.39/12.86 mark(isNatKind(x0)) 42.39/12.86 mark(U103(x0, x1, x2)) 42.39/12.86 mark(isNat(x0)) 42.39/12.86 mark(U104(x0, x1, x2)) 42.39/12.86 mark(plus(x0, x1)) 42.39/12.86 mark(x(x0, x1)) 42.39/12.86 mark(U11(x0, x1, x2)) 42.39/12.86 mark(U12(x0, x1, x2)) 42.39/12.86 mark(U13(x0, x1, x2)) 42.39/12.86 mark(U14(x0, x1, x2)) 42.39/12.86 mark(U15(x0, x1)) 42.39/12.86 mark(U16(x0)) 42.39/12.86 mark(U21(x0, x1)) 42.39/12.86 mark(U22(x0, x1)) 42.39/12.86 mark(U23(x0)) 42.39/12.86 mark(U31(x0, x1, x2)) 42.39/12.86 mark(U32(x0, x1, x2)) 42.39/12.86 mark(U33(x0, x1, x2)) 42.39/12.86 mark(U34(x0, x1, x2)) 42.39/12.86 mark(U35(x0, x1)) 42.39/12.86 mark(U36(x0)) 42.39/12.86 mark(U41(x0, x1)) 42.39/12.86 mark(U42(x0)) 42.39/12.86 mark(U51(x0)) 42.39/12.86 mark(U61(x0, x1)) 42.39/12.86 mark(U62(x0)) 42.39/12.86 mark(U71(x0, x1)) 42.39/12.86 mark(U72(x0, x1)) 42.39/12.86 mark(U81(x0, x1, x2)) 42.39/12.86 mark(U82(x0, x1, x2)) 42.39/12.86 mark(U83(x0, x1, x2)) 42.39/12.86 mark(U84(x0, x1, x2)) 42.39/12.86 mark(U91(x0, x1)) 42.39/12.86 mark(U92(x0)) 42.39/12.86 mark(tt) 42.39/12.86 mark(s(x0)) 42.39/12.86 mark(0) 42.39/12.86 a__U101(x0, x1, x2) 42.39/12.86 a__U102(x0, x1, x2) 42.39/12.86 a__isNatKind(x0) 42.39/12.86 a__U103(x0, x1, x2) 42.39/12.86 a__isNat(x0) 42.39/12.86 a__U104(x0, x1, x2) 42.39/12.86 a__plus(x0, x1) 42.39/12.86 a__x(x0, x1) 42.39/12.86 a__U11(x0, x1, x2) 42.39/12.86 a__U12(x0, x1, x2) 42.39/12.86 a__U13(x0, x1, x2) 42.39/12.86 a__U14(x0, x1, x2) 42.39/12.86 a__U15(x0, x1) 42.39/12.86 a__U16(x0) 42.39/12.86 a__U21(x0, x1) 42.39/12.86 a__U22(x0, x1) 42.39/12.86 a__U23(x0) 42.39/12.86 a__U31(x0, x1, x2) 42.39/12.86 a__U32(x0, x1, x2) 42.39/12.86 a__U33(x0, x1, x2) 42.39/12.86 a__U34(x0, x1, x2) 42.39/12.86 a__U35(x0, x1) 42.39/12.86 a__U36(x0) 42.39/12.86 a__U41(x0, x1) 42.39/12.86 a__U42(x0) 42.39/12.86 a__U51(x0) 42.39/12.86 a__U61(x0, x1) 42.39/12.86 a__U62(x0) 42.39/12.86 a__U71(x0, x1) 42.39/12.86 a__U72(x0, x1) 42.39/12.86 a__U81(x0, x1, x2) 42.39/12.86 a__U82(x0, x1, x2) 42.39/12.86 a__U83(x0, x1, x2) 42.39/12.86 a__U84(x0, x1, x2) 42.39/12.86 a__U91(x0, x1) 42.39/12.86 a__U92(x0) 42.39/12.86 42.39/12.86 We have to consider all minimal (P,Q,R)-chains. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (8) QReductionProof (EQUIVALENT) 42.39/12.86 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 42.39/12.86 42.39/12.86 mark(U101(x0, x1, x2)) 42.39/12.86 mark(U102(x0, x1, x2)) 42.39/12.86 mark(isNatKind(x0)) 42.39/12.86 mark(U103(x0, x1, x2)) 42.39/12.86 mark(isNat(x0)) 42.39/12.86 mark(U104(x0, x1, x2)) 42.39/12.86 mark(plus(x0, x1)) 42.39/12.86 mark(x(x0, x1)) 42.39/12.86 mark(U11(x0, x1, x2)) 42.39/12.86 mark(U12(x0, x1, x2)) 42.39/12.86 mark(U13(x0, x1, x2)) 42.39/12.86 mark(U14(x0, x1, x2)) 42.39/12.86 mark(U15(x0, x1)) 42.39/12.86 mark(U16(x0)) 42.39/12.86 mark(U21(x0, x1)) 42.39/12.86 mark(U22(x0, x1)) 42.39/12.86 mark(U23(x0)) 42.39/12.86 mark(U31(x0, x1, x2)) 42.39/12.86 mark(U32(x0, x1, x2)) 42.39/12.86 mark(U33(x0, x1, x2)) 42.39/12.86 mark(U34(x0, x1, x2)) 42.39/12.86 mark(U35(x0, x1)) 42.39/12.86 mark(U36(x0)) 42.39/12.86 mark(U41(x0, x1)) 42.39/12.86 mark(U42(x0)) 42.39/12.86 mark(U51(x0)) 42.39/12.86 mark(U61(x0, x1)) 42.39/12.86 mark(U62(x0)) 42.39/12.86 mark(U71(x0, x1)) 42.39/12.86 mark(U72(x0, x1)) 42.39/12.86 mark(U81(x0, x1, x2)) 42.39/12.86 mark(U82(x0, x1, x2)) 42.39/12.86 mark(U83(x0, x1, x2)) 42.39/12.86 mark(U84(x0, x1, x2)) 42.39/12.86 mark(U91(x0, x1)) 42.39/12.86 mark(U92(x0)) 42.39/12.86 mark(tt) 42.39/12.86 mark(s(x0)) 42.39/12.86 mark(0) 42.39/12.86 a__U101(x0, x1, x2) 42.39/12.86 a__U102(x0, x1, x2) 42.39/12.86 a__U103(x0, x1, x2) 42.39/12.86 a__isNat(x0) 42.39/12.86 a__U104(x0, x1, x2) 42.39/12.86 a__plus(x0, x1) 42.39/12.86 a__x(x0, x1) 42.39/12.86 a__U11(x0, x1, x2) 42.39/12.86 a__U12(x0, x1, x2) 42.39/12.86 a__U13(x0, x1, x2) 42.39/12.86 a__U14(x0, x1, x2) 42.39/12.86 a__U15(x0, x1) 42.39/12.86 a__U16(x0) 42.39/12.86 a__U21(x0, x1) 42.39/12.86 a__U22(x0, x1) 42.39/12.86 a__U23(x0) 42.39/12.86 a__U31(x0, x1, x2) 42.39/12.86 a__U32(x0, x1, x2) 42.39/12.86 a__U33(x0, x1, x2) 42.39/12.86 a__U34(x0, x1, x2) 42.39/12.86 a__U35(x0, x1) 42.39/12.86 a__U36(x0) 42.39/12.86 a__U71(x0, x1) 42.39/12.86 a__U72(x0, x1) 42.39/12.86 a__U81(x0, x1, x2) 42.39/12.86 a__U82(x0, x1, x2) 42.39/12.86 a__U83(x0, x1, x2) 42.39/12.86 a__U84(x0, x1, x2) 42.39/12.86 a__U91(x0, x1) 42.39/12.86 a__U92(x0) 42.39/12.86 42.39/12.86 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (9) 42.39/12.86 Obligation: 42.39/12.86 Q DP problem: 42.39/12.86 The TRS P consists of the following rules: 42.39/12.86 42.39/12.86 A__U41(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 A__ISNATKIND(plus(V1, V2)) -> A__U41(a__isNatKind(V1), V2) 42.39/12.86 A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.39/12.86 A__ISNATKIND(x(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.39/12.86 A__U61(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 A__ISNATKIND(x(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 42.39/12.86 The TRS R consists of the following rules: 42.39/12.86 42.39/12.86 a__isNatKind(0) -> tt 42.39/12.86 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.86 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(X) -> isNatKind(X) 42.39/12.86 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.86 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.86 a__U62(tt) -> tt 42.39/12.86 a__U62(X) -> U62(X) 42.39/12.86 a__U51(tt) -> tt 42.39/12.86 a__U51(X) -> U51(X) 42.39/12.86 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.86 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.86 a__U42(tt) -> tt 42.39/12.86 a__U42(X) -> U42(X) 42.39/12.86 42.39/12.86 The set Q consists of the following terms: 42.39/12.86 42.39/12.86 a__isNatKind(x0) 42.39/12.86 a__U41(x0, x1) 42.39/12.86 a__U42(x0) 42.39/12.86 a__U51(x0) 42.39/12.86 a__U61(x0, x1) 42.39/12.86 a__U62(x0) 42.39/12.86 42.39/12.86 We have to consider all minimal (P,Q,R)-chains. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (10) QDPSizeChangeProof (EQUIVALENT) 42.39/12.86 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. 42.39/12.86 42.39/12.86 From the DPs we obtained the following set of size-change graphs: 42.39/12.86 *A__ISNATKIND(plus(V1, V2)) -> A__U41(a__isNatKind(V1), V2) 42.39/12.86 The graph contains the following edges 1 > 2 42.39/12.86 42.39/12.86 42.39/12.86 *A__ISNATKIND(x(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.39/12.86 The graph contains the following edges 1 > 2 42.39/12.86 42.39/12.86 42.39/12.86 *A__U41(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 The graph contains the following edges 2 >= 1 42.39/12.86 42.39/12.86 42.39/12.86 *A__U61(tt, V2) -> A__ISNATKIND(V2) 42.39/12.86 The graph contains the following edges 2 >= 1 42.39/12.86 42.39/12.86 42.39/12.86 *A__ISNATKIND(plus(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 The graph contains the following edges 1 > 1 42.39/12.86 42.39/12.86 42.39/12.86 *A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.39/12.86 The graph contains the following edges 1 > 1 42.39/12.86 42.39/12.86 42.39/12.86 *A__ISNATKIND(x(V1, V2)) -> A__ISNATKIND(V1) 42.39/12.86 The graph contains the following edges 1 > 1 42.39/12.86 42.39/12.86 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (11) 42.39/12.86 YES 42.39/12.86 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (12) 42.39/12.86 Obligation: 42.39/12.86 Q DP problem: 42.39/12.86 The TRS P consists of the following rules: 42.39/12.86 42.39/12.86 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 42.39/12.86 A__U15(tt, V2) -> A__ISNAT(V2) 42.39/12.86 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 42.39/12.86 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 42.39/12.86 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.39/12.86 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.39/12.86 A__U22(tt, V1) -> A__ISNAT(V1) 42.39/12.86 A__ISNAT(x(V1, V2)) -> A__U31(a__isNatKind(V1), V1, V2) 42.39/12.86 A__U31(tt, V1, V2) -> A__U32(a__isNatKind(V1), V1, V2) 42.39/12.86 A__U32(tt, V1, V2) -> A__U33(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U33(tt, V1, V2) -> A__U34(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U34(tt, V1, V2) -> A__U35(a__isNat(V1), V2) 42.39/12.86 A__U35(tt, V2) -> A__ISNAT(V2) 42.39/12.86 A__U34(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.86 A__U14(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.86 42.39/12.86 The TRS R consists of the following rules: 42.39/12.86 42.39/12.86 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.86 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.86 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.86 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.86 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.86 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.86 a__U16(tt) -> tt 42.39/12.86 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.86 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.86 a__U23(tt) -> tt 42.39/12.86 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.86 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.86 a__U36(tt) -> tt 42.39/12.86 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.86 a__U42(tt) -> tt 42.39/12.86 a__U51(tt) -> tt 42.39/12.86 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.86 a__U62(tt) -> tt 42.39/12.86 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.86 a__U72(tt, N) -> mark(N) 42.39/12.86 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.86 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.86 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.86 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.86 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.86 a__U92(tt) -> 0 42.39/12.86 a__isNat(0) -> tt 42.39/12.86 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.86 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNatKind(0) -> tt 42.39/12.86 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.86 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.86 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.86 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.86 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.86 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.86 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.86 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.86 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.86 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.86 mark(isNat(X)) -> a__isNat(X) 42.39/12.86 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.86 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.86 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.86 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.86 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.86 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.86 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.86 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.86 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.86 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.86 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.86 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.86 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.86 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.86 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.86 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.86 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.86 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.86 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.86 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.86 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.86 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.86 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.86 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.86 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.86 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.86 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.86 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.86 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.86 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.86 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.86 mark(tt) -> tt 42.39/12.86 mark(s(X)) -> s(mark(X)) 42.39/12.86 mark(0) -> 0 42.39/12.86 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.86 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.86 a__isNatKind(X) -> isNatKind(X) 42.39/12.86 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.86 a__isNat(X) -> isNat(X) 42.39/12.86 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.86 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.86 a__x(X1, X2) -> x(X1, X2) 42.39/12.86 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.86 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.86 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.86 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.86 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.86 a__U16(X) -> U16(X) 42.39/12.86 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.86 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.86 a__U23(X) -> U23(X) 42.39/12.86 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.86 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.86 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.86 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.86 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.86 a__U36(X) -> U36(X) 42.39/12.86 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.86 a__U42(X) -> U42(X) 42.39/12.86 a__U51(X) -> U51(X) 42.39/12.86 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.86 a__U62(X) -> U62(X) 42.39/12.86 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.86 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.86 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.86 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.86 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.86 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.86 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.86 a__U92(X) -> U92(X) 42.39/12.86 42.39/12.86 The set Q consists of the following terms: 42.39/12.86 42.39/12.86 mark(U101(x0, x1, x2)) 42.39/12.86 mark(U102(x0, x1, x2)) 42.39/12.86 mark(isNatKind(x0)) 42.39/12.86 mark(U103(x0, x1, x2)) 42.39/12.86 mark(isNat(x0)) 42.39/12.86 mark(U104(x0, x1, x2)) 42.39/12.86 mark(plus(x0, x1)) 42.39/12.86 mark(x(x0, x1)) 42.39/12.86 mark(U11(x0, x1, x2)) 42.39/12.86 mark(U12(x0, x1, x2)) 42.39/12.86 mark(U13(x0, x1, x2)) 42.39/12.86 mark(U14(x0, x1, x2)) 42.39/12.86 mark(U15(x0, x1)) 42.39/12.86 mark(U16(x0)) 42.39/12.86 mark(U21(x0, x1)) 42.39/12.86 mark(U22(x0, x1)) 42.39/12.86 mark(U23(x0)) 42.39/12.86 mark(U31(x0, x1, x2)) 42.39/12.86 mark(U32(x0, x1, x2)) 42.39/12.86 mark(U33(x0, x1, x2)) 42.39/12.86 mark(U34(x0, x1, x2)) 42.39/12.86 mark(U35(x0, x1)) 42.39/12.86 mark(U36(x0)) 42.39/12.86 mark(U41(x0, x1)) 42.39/12.86 mark(U42(x0)) 42.39/12.86 mark(U51(x0)) 42.39/12.86 mark(U61(x0, x1)) 42.39/12.86 mark(U62(x0)) 42.39/12.86 mark(U71(x0, x1)) 42.39/12.86 mark(U72(x0, x1)) 42.39/12.86 mark(U81(x0, x1, x2)) 42.39/12.86 mark(U82(x0, x1, x2)) 42.39/12.86 mark(U83(x0, x1, x2)) 42.39/12.86 mark(U84(x0, x1, x2)) 42.39/12.86 mark(U91(x0, x1)) 42.39/12.86 mark(U92(x0)) 42.39/12.86 mark(tt) 42.39/12.86 mark(s(x0)) 42.39/12.86 mark(0) 42.39/12.86 a__U101(x0, x1, x2) 42.39/12.86 a__U102(x0, x1, x2) 42.39/12.86 a__isNatKind(x0) 42.39/12.86 a__U103(x0, x1, x2) 42.39/12.86 a__isNat(x0) 42.39/12.86 a__U104(x0, x1, x2) 42.39/12.86 a__plus(x0, x1) 42.39/12.86 a__x(x0, x1) 42.39/12.86 a__U11(x0, x1, x2) 42.39/12.86 a__U12(x0, x1, x2) 42.39/12.86 a__U13(x0, x1, x2) 42.39/12.86 a__U14(x0, x1, x2) 42.39/12.86 a__U15(x0, x1) 42.39/12.86 a__U16(x0) 42.39/12.86 a__U21(x0, x1) 42.39/12.86 a__U22(x0, x1) 42.39/12.86 a__U23(x0) 42.39/12.86 a__U31(x0, x1, x2) 42.39/12.86 a__U32(x0, x1, x2) 42.39/12.86 a__U33(x0, x1, x2) 42.39/12.86 a__U34(x0, x1, x2) 42.39/12.86 a__U35(x0, x1) 42.39/12.86 a__U36(x0) 42.39/12.86 a__U41(x0, x1) 42.39/12.86 a__U42(x0) 42.39/12.86 a__U51(x0) 42.39/12.86 a__U61(x0, x1) 42.39/12.86 a__U62(x0) 42.39/12.86 a__U71(x0, x1) 42.39/12.86 a__U72(x0, x1) 42.39/12.86 a__U81(x0, x1, x2) 42.39/12.86 a__U82(x0, x1, x2) 42.39/12.86 a__U83(x0, x1, x2) 42.39/12.86 a__U84(x0, x1, x2) 42.39/12.86 a__U91(x0, x1) 42.39/12.86 a__U92(x0) 42.39/12.86 42.39/12.86 We have to consider all minimal (P,Q,R)-chains. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (13) UsableRulesProof (EQUIVALENT) 42.39/12.86 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. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (14) 42.39/12.86 Obligation: 42.39/12.86 Q DP problem: 42.39/12.86 The TRS P consists of the following rules: 42.39/12.86 42.39/12.86 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 42.39/12.86 A__U15(tt, V2) -> A__ISNAT(V2) 42.39/12.86 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 42.39/12.86 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 42.39/12.86 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.39/12.86 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.39/12.86 A__U22(tt, V1) -> A__ISNAT(V1) 42.39/12.86 A__ISNAT(x(V1, V2)) -> A__U31(a__isNatKind(V1), V1, V2) 42.39/12.86 A__U31(tt, V1, V2) -> A__U32(a__isNatKind(V1), V1, V2) 42.39/12.86 A__U32(tt, V1, V2) -> A__U33(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U33(tt, V1, V2) -> A__U34(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U34(tt, V1, V2) -> A__U35(a__isNat(V1), V2) 42.39/12.86 A__U35(tt, V2) -> A__ISNAT(V2) 42.39/12.86 A__U34(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.86 A__U14(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.86 42.39/12.86 The TRS R consists of the following rules: 42.39/12.86 42.39/12.86 a__isNat(0) -> tt 42.39/12.86 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.86 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.86 a__isNat(X) -> isNat(X) 42.39/12.86 a__isNatKind(0) -> tt 42.39/12.86 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.86 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.86 a__isNatKind(X) -> isNatKind(X) 42.39/12.86 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.86 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.86 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.86 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.86 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.86 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.86 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.86 a__U36(tt) -> tt 42.39/12.86 a__U36(X) -> U36(X) 42.39/12.86 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.86 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.86 a__U62(tt) -> tt 42.39/12.86 a__U62(X) -> U62(X) 42.39/12.86 a__U51(tt) -> tt 42.39/12.86 a__U51(X) -> U51(X) 42.39/12.86 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.86 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.86 a__U42(tt) -> tt 42.39/12.86 a__U42(X) -> U42(X) 42.39/12.86 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.86 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.86 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.86 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.86 a__U23(tt) -> tt 42.39/12.86 a__U23(X) -> U23(X) 42.39/12.86 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.86 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.86 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.86 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.86 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.86 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.86 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.86 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.86 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.86 a__U16(tt) -> tt 42.39/12.86 a__U16(X) -> U16(X) 42.39/12.86 42.39/12.86 The set Q consists of the following terms: 42.39/12.86 42.39/12.86 mark(U101(x0, x1, x2)) 42.39/12.86 mark(U102(x0, x1, x2)) 42.39/12.86 mark(isNatKind(x0)) 42.39/12.86 mark(U103(x0, x1, x2)) 42.39/12.86 mark(isNat(x0)) 42.39/12.86 mark(U104(x0, x1, x2)) 42.39/12.86 mark(plus(x0, x1)) 42.39/12.86 mark(x(x0, x1)) 42.39/12.86 mark(U11(x0, x1, x2)) 42.39/12.86 mark(U12(x0, x1, x2)) 42.39/12.86 mark(U13(x0, x1, x2)) 42.39/12.86 mark(U14(x0, x1, x2)) 42.39/12.86 mark(U15(x0, x1)) 42.39/12.86 mark(U16(x0)) 42.39/12.86 mark(U21(x0, x1)) 42.39/12.86 mark(U22(x0, x1)) 42.39/12.86 mark(U23(x0)) 42.39/12.86 mark(U31(x0, x1, x2)) 42.39/12.86 mark(U32(x0, x1, x2)) 42.39/12.86 mark(U33(x0, x1, x2)) 42.39/12.86 mark(U34(x0, x1, x2)) 42.39/12.86 mark(U35(x0, x1)) 42.39/12.86 mark(U36(x0)) 42.39/12.86 mark(U41(x0, x1)) 42.39/12.86 mark(U42(x0)) 42.39/12.86 mark(U51(x0)) 42.39/12.86 mark(U61(x0, x1)) 42.39/12.86 mark(U62(x0)) 42.39/12.86 mark(U71(x0, x1)) 42.39/12.86 mark(U72(x0, x1)) 42.39/12.86 mark(U81(x0, x1, x2)) 42.39/12.86 mark(U82(x0, x1, x2)) 42.39/12.86 mark(U83(x0, x1, x2)) 42.39/12.86 mark(U84(x0, x1, x2)) 42.39/12.86 mark(U91(x0, x1)) 42.39/12.86 mark(U92(x0)) 42.39/12.86 mark(tt) 42.39/12.86 mark(s(x0)) 42.39/12.86 mark(0) 42.39/12.86 a__U101(x0, x1, x2) 42.39/12.86 a__U102(x0, x1, x2) 42.39/12.86 a__isNatKind(x0) 42.39/12.86 a__U103(x0, x1, x2) 42.39/12.86 a__isNat(x0) 42.39/12.86 a__U104(x0, x1, x2) 42.39/12.86 a__plus(x0, x1) 42.39/12.86 a__x(x0, x1) 42.39/12.86 a__U11(x0, x1, x2) 42.39/12.86 a__U12(x0, x1, x2) 42.39/12.86 a__U13(x0, x1, x2) 42.39/12.86 a__U14(x0, x1, x2) 42.39/12.86 a__U15(x0, x1) 42.39/12.86 a__U16(x0) 42.39/12.86 a__U21(x0, x1) 42.39/12.86 a__U22(x0, x1) 42.39/12.86 a__U23(x0) 42.39/12.86 a__U31(x0, x1, x2) 42.39/12.86 a__U32(x0, x1, x2) 42.39/12.86 a__U33(x0, x1, x2) 42.39/12.86 a__U34(x0, x1, x2) 42.39/12.86 a__U35(x0, x1) 42.39/12.86 a__U36(x0) 42.39/12.86 a__U41(x0, x1) 42.39/12.86 a__U42(x0) 42.39/12.86 a__U51(x0) 42.39/12.86 a__U61(x0, x1) 42.39/12.86 a__U62(x0) 42.39/12.86 a__U71(x0, x1) 42.39/12.86 a__U72(x0, x1) 42.39/12.86 a__U81(x0, x1, x2) 42.39/12.86 a__U82(x0, x1, x2) 42.39/12.86 a__U83(x0, x1, x2) 42.39/12.86 a__U84(x0, x1, x2) 42.39/12.86 a__U91(x0, x1) 42.39/12.86 a__U92(x0) 42.39/12.86 42.39/12.86 We have to consider all minimal (P,Q,R)-chains. 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (15) QReductionProof (EQUIVALENT) 42.39/12.86 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 42.39/12.86 42.39/12.86 mark(U101(x0, x1, x2)) 42.39/12.86 mark(U102(x0, x1, x2)) 42.39/12.86 mark(isNatKind(x0)) 42.39/12.86 mark(U103(x0, x1, x2)) 42.39/12.86 mark(isNat(x0)) 42.39/12.86 mark(U104(x0, x1, x2)) 42.39/12.86 mark(plus(x0, x1)) 42.39/12.86 mark(x(x0, x1)) 42.39/12.86 mark(U11(x0, x1, x2)) 42.39/12.86 mark(U12(x0, x1, x2)) 42.39/12.86 mark(U13(x0, x1, x2)) 42.39/12.86 mark(U14(x0, x1, x2)) 42.39/12.86 mark(U15(x0, x1)) 42.39/12.86 mark(U16(x0)) 42.39/12.86 mark(U21(x0, x1)) 42.39/12.86 mark(U22(x0, x1)) 42.39/12.86 mark(U23(x0)) 42.39/12.86 mark(U31(x0, x1, x2)) 42.39/12.86 mark(U32(x0, x1, x2)) 42.39/12.86 mark(U33(x0, x1, x2)) 42.39/12.86 mark(U34(x0, x1, x2)) 42.39/12.86 mark(U35(x0, x1)) 42.39/12.86 mark(U36(x0)) 42.39/12.86 mark(U41(x0, x1)) 42.39/12.86 mark(U42(x0)) 42.39/12.86 mark(U51(x0)) 42.39/12.86 mark(U61(x0, x1)) 42.39/12.86 mark(U62(x0)) 42.39/12.86 mark(U71(x0, x1)) 42.39/12.86 mark(U72(x0, x1)) 42.39/12.86 mark(U81(x0, x1, x2)) 42.39/12.86 mark(U82(x0, x1, x2)) 42.39/12.86 mark(U83(x0, x1, x2)) 42.39/12.86 mark(U84(x0, x1, x2)) 42.39/12.86 mark(U91(x0, x1)) 42.39/12.86 mark(U92(x0)) 42.39/12.86 mark(tt) 42.39/12.86 mark(s(x0)) 42.39/12.86 mark(0) 42.39/12.86 a__U101(x0, x1, x2) 42.39/12.86 a__U102(x0, x1, x2) 42.39/12.86 a__U103(x0, x1, x2) 42.39/12.86 a__U104(x0, x1, x2) 42.39/12.86 a__plus(x0, x1) 42.39/12.86 a__x(x0, x1) 42.39/12.86 a__U71(x0, x1) 42.39/12.86 a__U72(x0, x1) 42.39/12.86 a__U81(x0, x1, x2) 42.39/12.86 a__U82(x0, x1, x2) 42.39/12.86 a__U83(x0, x1, x2) 42.39/12.86 a__U84(x0, x1, x2) 42.39/12.86 a__U91(x0, x1) 42.39/12.86 a__U92(x0) 42.39/12.86 42.39/12.86 42.39/12.86 ---------------------------------------- 42.39/12.86 42.39/12.86 (16) 42.39/12.86 Obligation: 42.39/12.86 Q DP problem: 42.39/12.86 The TRS P consists of the following rules: 42.39/12.86 42.39/12.86 A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 42.39/12.86 A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 42.39/12.86 A__U15(tt, V2) -> A__ISNAT(V2) 42.39/12.86 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 42.39/12.86 A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 42.39/12.86 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.39/12.86 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.39/12.86 A__U22(tt, V1) -> A__ISNAT(V1) 42.39/12.86 A__ISNAT(x(V1, V2)) -> A__U31(a__isNatKind(V1), V1, V2) 42.39/12.87 A__U31(tt, V1, V2) -> A__U32(a__isNatKind(V1), V1, V2) 42.39/12.87 A__U32(tt, V1, V2) -> A__U33(a__isNatKind(V2), V1, V2) 42.39/12.87 A__U33(tt, V1, V2) -> A__U34(a__isNatKind(V2), V1, V2) 42.39/12.87 A__U34(tt, V1, V2) -> A__U35(a__isNat(V1), V2) 42.39/12.87 A__U35(tt, V2) -> A__ISNAT(V2) 42.39/12.87 A__U34(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.87 A__U14(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.87 42.39/12.87 The TRS R consists of the following rules: 42.39/12.87 42.39/12.87 a__isNat(0) -> tt 42.39/12.87 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.87 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.87 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.87 a__isNat(X) -> isNat(X) 42.39/12.87 a__isNatKind(0) -> tt 42.39/12.87 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.87 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.87 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.87 a__isNatKind(X) -> isNatKind(X) 42.39/12.87 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.87 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.87 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.87 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.87 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.87 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.87 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.87 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.87 a__U36(tt) -> tt 42.39/12.87 a__U36(X) -> U36(X) 42.39/12.87 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.87 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.87 a__U62(tt) -> tt 42.39/12.87 a__U62(X) -> U62(X) 42.39/12.87 a__U51(tt) -> tt 42.39/12.87 a__U51(X) -> U51(X) 42.39/12.87 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.87 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.87 a__U42(tt) -> tt 42.39/12.87 a__U42(X) -> U42(X) 42.39/12.87 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.87 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.87 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.87 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.87 a__U23(tt) -> tt 42.39/12.87 a__U23(X) -> U23(X) 42.39/12.87 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.87 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.87 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.87 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.87 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.87 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.87 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.87 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.87 a__U16(tt) -> tt 42.39/12.87 a__U16(X) -> U16(X) 42.39/12.87 42.39/12.87 The set Q consists of the following terms: 42.39/12.87 42.39/12.87 a__isNatKind(x0) 42.39/12.87 a__isNat(x0) 42.39/12.87 a__U11(x0, x1, x2) 42.39/12.87 a__U12(x0, x1, x2) 42.39/12.87 a__U13(x0, x1, x2) 42.39/12.87 a__U14(x0, x1, x2) 42.39/12.87 a__U15(x0, x1) 42.39/12.87 a__U16(x0) 42.39/12.87 a__U21(x0, x1) 42.39/12.87 a__U22(x0, x1) 42.39/12.87 a__U23(x0) 42.39/12.87 a__U31(x0, x1, x2) 42.39/12.87 a__U32(x0, x1, x2) 42.39/12.87 a__U33(x0, x1, x2) 42.39/12.87 a__U34(x0, x1, x2) 42.39/12.87 a__U35(x0, x1) 42.39/12.87 a__U36(x0) 42.39/12.87 a__U41(x0, x1) 42.39/12.87 a__U42(x0) 42.39/12.87 a__U51(x0) 42.39/12.87 a__U61(x0, x1) 42.39/12.87 a__U62(x0) 42.39/12.87 42.39/12.87 We have to consider all minimal (P,Q,R)-chains. 42.39/12.87 ---------------------------------------- 42.39/12.87 42.39/12.87 (17) QDPSizeChangeProof (EQUIVALENT) 42.39/12.87 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. 42.39/12.87 42.39/12.87 From the DPs we obtained the following set of size-change graphs: 42.39/12.87 *A__U13(tt, V1, V2) -> A__U14(a__isNatKind(V2), V1, V2) 42.39/12.87 The graph contains the following edges 2 >= 2, 3 >= 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U11(tt, V1, V2) -> A__U12(a__isNatKind(V1), V1, V2) 42.39/12.87 The graph contains the following edges 2 >= 2, 3 >= 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U12(tt, V1, V2) -> A__U13(a__isNatKind(V2), V1, V2) 42.39/12.87 The graph contains the following edges 2 >= 2, 3 >= 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U15(tt, V2) -> A__ISNAT(V2) 42.39/12.87 The graph contains the following edges 2 >= 1 42.39/12.87 42.39/12.87 42.39/12.87 *A__U14(tt, V1, V2) -> A__U15(a__isNat(V1), V2) 42.39/12.87 The graph contains the following edges 3 >= 2 42.39/12.87 42.39/12.87 42.39/12.87 *A__U14(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.87 The graph contains the following edges 2 >= 1 42.39/12.87 42.39/12.87 42.39/12.87 *A__ISNAT(plus(V1, V2)) -> A__U11(a__isNatKind(V1), V1, V2) 42.39/12.87 The graph contains the following edges 1 > 2, 1 > 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.39/12.87 The graph contains the following edges 2 >= 2 42.39/12.87 42.39/12.87 42.39/12.87 *A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.39/12.87 The graph contains the following edges 1 > 2 42.39/12.87 42.39/12.87 42.39/12.87 *A__ISNAT(x(V1, V2)) -> A__U31(a__isNatKind(V1), V1, V2) 42.39/12.87 The graph contains the following edges 1 > 2, 1 > 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U22(tt, V1) -> A__ISNAT(V1) 42.39/12.87 The graph contains the following edges 2 >= 1 42.39/12.87 42.39/12.87 42.39/12.87 *A__U31(tt, V1, V2) -> A__U32(a__isNatKind(V1), V1, V2) 42.39/12.87 The graph contains the following edges 2 >= 2, 3 >= 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U32(tt, V1, V2) -> A__U33(a__isNatKind(V2), V1, V2) 42.39/12.87 The graph contains the following edges 2 >= 2, 3 >= 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U33(tt, V1, V2) -> A__U34(a__isNatKind(V2), V1, V2) 42.39/12.87 The graph contains the following edges 2 >= 2, 3 >= 3 42.39/12.87 42.39/12.87 42.39/12.87 *A__U34(tt, V1, V2) -> A__U35(a__isNat(V1), V2) 42.39/12.87 The graph contains the following edges 3 >= 2 42.39/12.87 42.39/12.87 42.39/12.87 *A__U34(tt, V1, V2) -> A__ISNAT(V1) 42.39/12.87 The graph contains the following edges 2 >= 1 42.39/12.87 42.39/12.87 42.39/12.87 *A__U35(tt, V2) -> A__ISNAT(V2) 42.39/12.87 The graph contains the following edges 2 >= 1 42.39/12.87 42.39/12.87 42.39/12.87 ---------------------------------------- 42.39/12.87 42.39/12.87 (18) 42.39/12.87 YES 42.39/12.87 42.39/12.87 ---------------------------------------- 42.39/12.87 42.39/12.87 (19) 42.39/12.87 Obligation: 42.39/12.87 Q DP problem: 42.39/12.87 The TRS P consists of the following rules: 42.39/12.87 42.39/12.87 A__U102(tt, M, N) -> A__U103(a__isNat(N), M, N) 42.39/12.87 A__U103(tt, M, N) -> A__U104(a__isNatKind(N), M, N) 42.39/12.87 A__U104(tt, M, N) -> A__PLUS(a__x(mark(N), mark(M)), mark(N)) 42.39/12.87 A__PLUS(N, 0) -> A__U71(a__isNat(N), N) 42.39/12.87 A__U71(tt, N) -> A__U72(a__isNatKind(N), N) 42.39/12.87 A__U72(tt, N) -> MARK(N) 42.39/12.87 MARK(U101(X1, X2, X3)) -> A__U101(mark(X1), X2, X3) 42.39/12.87 A__U101(tt, M, N) -> A__U102(a__isNatKind(M), M, N) 42.39/12.87 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U102(X1, X2, X3)) -> A__U102(mark(X1), X2, X3) 42.39/12.87 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U103(X1, X2, X3)) -> A__U103(mark(X1), X2, X3) 42.39/12.87 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U104(X1, X2, X3)) -> A__U104(mark(X1), X2, X3) 42.39/12.87 A__U104(tt, M, N) -> A__X(mark(N), mark(M)) 42.39/12.87 A__X(N, s(M)) -> A__U101(a__isNat(M), M, N) 42.39/12.87 A__U104(tt, M, N) -> MARK(N) 42.39/12.87 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 42.39/12.87 A__PLUS(N, s(M)) -> A__U81(a__isNat(M), M, N) 42.39/12.87 A__U81(tt, M, N) -> A__U82(a__isNatKind(M), M, N) 42.39/12.87 A__U82(tt, M, N) -> A__U83(a__isNat(N), M, N) 42.39/12.87 A__U83(tt, M, N) -> A__U84(a__isNatKind(N), M, N) 42.39/12.87 A__U84(tt, M, N) -> A__PLUS(mark(N), mark(M)) 42.39/12.87 A__U84(tt, M, N) -> MARK(N) 42.39/12.87 MARK(plus(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(plus(X1, X2)) -> MARK(X2) 42.39/12.87 MARK(x(X1, X2)) -> A__X(mark(X1), mark(X2)) 42.39/12.87 MARK(x(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(x(X1, X2)) -> MARK(X2) 42.39/12.87 MARK(U11(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U12(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U13(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U14(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U15(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U16(X)) -> MARK(X) 42.39/12.87 MARK(U21(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U22(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U23(X)) -> MARK(X) 42.39/12.87 MARK(U31(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U32(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U33(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U34(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U35(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U36(X)) -> MARK(X) 42.39/12.87 MARK(U41(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U42(X)) -> MARK(X) 42.39/12.87 MARK(U51(X)) -> MARK(X) 42.39/12.87 MARK(U61(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U62(X)) -> MARK(X) 42.39/12.87 MARK(U71(X1, X2)) -> A__U71(mark(X1), X2) 42.39/12.87 MARK(U71(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U72(X1, X2)) -> A__U72(mark(X1), X2) 42.39/12.87 MARK(U72(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U81(X1, X2, X3)) -> A__U81(mark(X1), X2, X3) 42.39/12.87 MARK(U81(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U82(X1, X2, X3)) -> A__U82(mark(X1), X2, X3) 42.39/12.87 MARK(U82(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U83(X1, X2, X3)) -> A__U83(mark(X1), X2, X3) 42.39/12.87 MARK(U83(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U84(X1, X2, X3)) -> A__U84(mark(X1), X2, X3) 42.39/12.87 A__U84(tt, M, N) -> MARK(M) 42.39/12.87 MARK(U84(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U91(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U92(X)) -> MARK(X) 42.39/12.87 MARK(s(X)) -> MARK(X) 42.39/12.87 A__U104(tt, M, N) -> MARK(M) 42.39/12.87 42.39/12.87 The TRS R consists of the following rules: 42.39/12.87 42.39/12.87 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.87 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.87 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.87 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.87 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.87 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.87 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.87 a__U16(tt) -> tt 42.39/12.87 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.87 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.87 a__U23(tt) -> tt 42.39/12.87 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.87 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.87 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.87 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.87 a__U36(tt) -> tt 42.39/12.87 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.87 a__U42(tt) -> tt 42.39/12.87 a__U51(tt) -> tt 42.39/12.87 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.87 a__U62(tt) -> tt 42.39/12.87 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.87 a__U72(tt, N) -> mark(N) 42.39/12.87 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.87 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.87 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.87 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.87 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.87 a__U92(tt) -> 0 42.39/12.87 a__isNat(0) -> tt 42.39/12.87 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.87 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.87 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.87 a__isNatKind(0) -> tt 42.39/12.87 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.87 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.87 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.87 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.87 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.87 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.87 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.87 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.87 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.87 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.87 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.87 mark(isNat(X)) -> a__isNat(X) 42.39/12.87 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.87 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.87 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.87 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.87 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.87 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.87 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.87 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.87 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.87 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.87 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.87 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.87 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.87 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.87 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.87 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.87 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.87 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.87 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.87 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.87 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.87 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.87 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.87 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.87 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.87 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.87 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.87 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.87 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.87 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.87 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.87 mark(tt) -> tt 42.39/12.87 mark(s(X)) -> s(mark(X)) 42.39/12.87 mark(0) -> 0 42.39/12.87 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.87 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.87 a__isNatKind(X) -> isNatKind(X) 42.39/12.87 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.87 a__isNat(X) -> isNat(X) 42.39/12.87 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.87 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.87 a__x(X1, X2) -> x(X1, X2) 42.39/12.87 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.87 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.87 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.87 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.87 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.87 a__U16(X) -> U16(X) 42.39/12.87 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.87 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.87 a__U23(X) -> U23(X) 42.39/12.87 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.87 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.87 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.87 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.87 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.87 a__U36(X) -> U36(X) 42.39/12.87 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.87 a__U42(X) -> U42(X) 42.39/12.87 a__U51(X) -> U51(X) 42.39/12.87 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.87 a__U62(X) -> U62(X) 42.39/12.87 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.87 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.87 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.87 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.87 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.87 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.87 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.87 a__U92(X) -> U92(X) 42.39/12.87 42.39/12.87 The set Q consists of the following terms: 42.39/12.87 42.39/12.87 mark(U101(x0, x1, x2)) 42.39/12.87 mark(U102(x0, x1, x2)) 42.39/12.87 mark(isNatKind(x0)) 42.39/12.87 mark(U103(x0, x1, x2)) 42.39/12.87 mark(isNat(x0)) 42.39/12.87 mark(U104(x0, x1, x2)) 42.39/12.87 mark(plus(x0, x1)) 42.39/12.87 mark(x(x0, x1)) 42.39/12.87 mark(U11(x0, x1, x2)) 42.39/12.87 mark(U12(x0, x1, x2)) 42.39/12.87 mark(U13(x0, x1, x2)) 42.39/12.87 mark(U14(x0, x1, x2)) 42.39/12.87 mark(U15(x0, x1)) 42.39/12.87 mark(U16(x0)) 42.39/12.87 mark(U21(x0, x1)) 42.39/12.87 mark(U22(x0, x1)) 42.39/12.87 mark(U23(x0)) 42.39/12.87 mark(U31(x0, x1, x2)) 42.39/12.87 mark(U32(x0, x1, x2)) 42.39/12.87 mark(U33(x0, x1, x2)) 42.39/12.87 mark(U34(x0, x1, x2)) 42.39/12.87 mark(U35(x0, x1)) 42.39/12.87 mark(U36(x0)) 42.39/12.87 mark(U41(x0, x1)) 42.39/12.87 mark(U42(x0)) 42.39/12.87 mark(U51(x0)) 42.39/12.87 mark(U61(x0, x1)) 42.39/12.87 mark(U62(x0)) 42.39/12.87 mark(U71(x0, x1)) 42.39/12.87 mark(U72(x0, x1)) 42.39/12.87 mark(U81(x0, x1, x2)) 42.39/12.87 mark(U82(x0, x1, x2)) 42.39/12.87 mark(U83(x0, x1, x2)) 42.39/12.87 mark(U84(x0, x1, x2)) 42.39/12.87 mark(U91(x0, x1)) 42.39/12.87 mark(U92(x0)) 42.39/12.87 mark(tt) 42.39/12.87 mark(s(x0)) 42.39/12.87 mark(0) 42.39/12.87 a__U101(x0, x1, x2) 42.39/12.87 a__U102(x0, x1, x2) 42.39/12.87 a__isNatKind(x0) 42.39/12.87 a__U103(x0, x1, x2) 42.39/12.87 a__isNat(x0) 42.39/12.87 a__U104(x0, x1, x2) 42.39/12.87 a__plus(x0, x1) 42.39/12.87 a__x(x0, x1) 42.39/12.87 a__U11(x0, x1, x2) 42.39/12.87 a__U12(x0, x1, x2) 42.39/12.87 a__U13(x0, x1, x2) 42.39/12.87 a__U14(x0, x1, x2) 42.39/12.87 a__U15(x0, x1) 42.39/12.87 a__U16(x0) 42.39/12.87 a__U21(x0, x1) 42.39/12.87 a__U22(x0, x1) 42.39/12.87 a__U23(x0) 42.39/12.87 a__U31(x0, x1, x2) 42.39/12.87 a__U32(x0, x1, x2) 42.39/12.87 a__U33(x0, x1, x2) 42.39/12.87 a__U34(x0, x1, x2) 42.39/12.87 a__U35(x0, x1) 42.39/12.87 a__U36(x0) 42.39/12.87 a__U41(x0, x1) 42.39/12.87 a__U42(x0) 42.39/12.87 a__U51(x0) 42.39/12.87 a__U61(x0, x1) 42.39/12.87 a__U62(x0) 42.39/12.87 a__U71(x0, x1) 42.39/12.87 a__U72(x0, x1) 42.39/12.87 a__U81(x0, x1, x2) 42.39/12.87 a__U82(x0, x1, x2) 42.39/12.87 a__U83(x0, x1, x2) 42.39/12.87 a__U84(x0, x1, x2) 42.39/12.87 a__U91(x0, x1) 42.39/12.87 a__U92(x0) 42.39/12.87 42.39/12.87 We have to consider all minimal (P,Q,R)-chains. 42.39/12.87 ---------------------------------------- 42.39/12.87 42.39/12.87 (20) QDPOrderProof (EQUIVALENT) 42.39/12.87 We use the reduction pair processor [LPAR04,JAR06]. 42.39/12.87 42.39/12.87 42.39/12.87 The following pairs can be oriented strictly and are deleted. 42.39/12.87 42.39/12.87 A__U104(tt, M, N) -> A__PLUS(a__x(mark(N), mark(M)), mark(N)) 42.39/12.87 A__PLUS(N, 0) -> A__U71(a__isNat(N), N) 42.39/12.87 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.39/12.87 A__U104(tt, M, N) -> A__X(mark(N), mark(M)) 42.39/12.87 A__X(N, s(M)) -> A__U101(a__isNat(M), M, N) 42.39/12.87 A__U104(tt, M, N) -> MARK(N) 42.39/12.87 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 42.39/12.87 A__PLUS(N, s(M)) -> A__U81(a__isNat(M), M, N) 42.39/12.87 A__U83(tt, M, N) -> A__U84(a__isNatKind(N), M, N) 42.39/12.87 A__U84(tt, M, N) -> MARK(N) 42.39/12.87 MARK(plus(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(plus(X1, X2)) -> MARK(X2) 42.39/12.87 MARK(x(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(x(X1, X2)) -> MARK(X2) 42.39/12.87 MARK(U71(X1, X2)) -> A__U71(mark(X1), X2) 42.39/12.87 MARK(U71(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U72(X1, X2)) -> A__U72(mark(X1), X2) 42.39/12.87 MARK(U72(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(U81(X1, X2, X3)) -> A__U81(mark(X1), X2, X3) 42.39/12.87 MARK(U81(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U82(X1, X2, X3)) -> A__U82(mark(X1), X2, X3) 42.39/12.87 MARK(U82(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U83(X1, X2, X3)) -> A__U83(mark(X1), X2, X3) 42.39/12.87 MARK(U83(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U84(X1, X2, X3)) -> A__U84(mark(X1), X2, X3) 42.39/12.87 A__U84(tt, M, N) -> MARK(M) 42.39/12.87 MARK(U84(X1, X2, X3)) -> MARK(X1) 42.39/12.87 MARK(U91(X1, X2)) -> MARK(X1) 42.39/12.87 MARK(s(X)) -> MARK(X) 42.39/12.87 A__U104(tt, M, N) -> MARK(M) 42.39/12.87 The remaining pairs can at least be oriented weakly. 42.39/12.87 Used ordering: Combined order from the following AFS and order. 42.39/12.87 A__U102(x1, x2, x3) = A__U102(x1, x2, x3) 42.39/12.87 42.39/12.87 tt = tt 42.39/12.87 42.39/12.87 A__U103(x1, x2, x3) = A__U103(x1, x2, x3) 42.39/12.87 42.39/12.87 a__isNat(x1) = a__isNat 42.39/12.87 42.39/12.87 A__U104(x1, x2, x3) = A__U104(x1, x2, x3) 42.39/12.87 42.39/12.87 a__isNatKind(x1) = a__isNatKind 42.39/12.87 42.39/12.87 A__PLUS(x1, x2) = A__PLUS(x1, x2) 42.39/12.87 42.39/12.87 a__x(x1, x2) = a__x(x1, x2) 42.39/12.87 42.39/12.87 mark(x1) = x1 42.39/12.87 42.39/12.87 0 = 0 42.39/12.87 42.39/12.87 A__U71(x1, x2) = x2 42.39/12.87 42.39/12.87 A__U72(x1, x2) = x2 42.39/12.87 42.39/12.87 MARK(x1) = x1 42.39/12.87 42.39/12.87 U101(x1, x2, x3) = U101(x1, x2, x3) 42.39/12.88 42.39/12.88 A__U101(x1, x2, x3) = A__U101(x1, x2, x3) 42.39/12.88 42.39/12.88 U102(x1, x2, x3) = U102(x1, x2, x3) 42.39/12.88 42.39/12.88 U103(x1, x2, x3) = U103(x1, x2, x3) 42.39/12.88 42.39/12.88 U104(x1, x2, x3) = U104(x1, x2, x3) 42.39/12.88 42.39/12.88 A__X(x1, x2) = A__X(x1, x2) 42.39/12.88 42.39/12.88 s(x1) = s(x1) 42.39/12.88 42.39/12.88 plus(x1, x2) = plus(x1, x2) 42.39/12.88 42.39/12.88 A__U81(x1, x2, x3) = A__U81(x1, x2, x3) 42.39/12.88 42.39/12.88 A__U82(x1, x2, x3) = A__U82(x1, x2, x3) 42.39/12.88 42.39/12.88 A__U83(x1, x2, x3) = A__U83(x1, x2, x3) 42.39/12.88 42.39/12.88 A__U84(x1, x2, x3) = A__U84(x2, x3) 42.39/12.88 42.39/12.88 x(x1, x2) = x(x1, x2) 42.39/12.88 42.39/12.88 U11(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U12(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U13(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U14(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U15(x1, x2) = x1 42.39/12.88 42.39/12.88 U16(x1) = x1 42.39/12.88 42.39/12.88 U21(x1, x2) = x1 42.39/12.88 42.39/12.88 U22(x1, x2) = x1 42.39/12.88 42.39/12.88 U23(x1) = x1 42.39/12.88 42.39/12.88 U31(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U32(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U33(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U34(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 U35(x1, x2) = x1 42.39/12.88 42.39/12.88 U36(x1) = x1 42.39/12.88 42.39/12.88 U41(x1, x2) = x1 42.39/12.88 42.39/12.88 U42(x1) = x1 42.39/12.88 42.39/12.88 U51(x1) = x1 42.39/12.88 42.39/12.88 U61(x1, x2) = x1 42.39/12.88 42.39/12.88 U62(x1) = x1 42.39/12.88 42.39/12.88 U71(x1, x2) = U71(x1, x2) 42.39/12.88 42.39/12.88 U72(x1, x2) = U72(x1, x2) 42.39/12.88 42.39/12.88 U81(x1, x2, x3) = U81(x1, x2, x3) 42.39/12.88 42.39/12.88 U82(x1, x2, x3) = U82(x1, x2, x3) 42.39/12.88 42.39/12.88 U83(x1, x2, x3) = U83(x1, x2, x3) 42.39/12.88 42.39/12.88 U84(x1, x2, x3) = U84(x1, x2, x3) 42.39/12.88 42.39/12.88 U91(x1, x2) = U91(x1, x2) 42.39/12.88 42.39/12.88 U92(x1) = x1 42.39/12.88 42.39/12.88 a__U11(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 a__U21(x1, x2) = x1 42.39/12.88 42.39/12.88 a__U31(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 isNat(x1) = isNat 42.39/12.88 42.39/12.88 a__U41(x1, x2) = x1 42.39/12.88 42.39/12.88 a__U51(x1) = x1 42.39/12.88 42.39/12.88 a__U61(x1, x2) = x1 42.39/12.88 42.39/12.88 isNatKind(x1) = isNatKind 42.39/12.88 42.39/12.88 a__U102(x1, x2, x3) = a__U102(x1, x2, x3) 42.39/12.88 42.39/12.88 a__U103(x1, x2, x3) = a__U103(x1, x2, x3) 42.39/12.88 42.39/12.88 a__U104(x1, x2, x3) = a__U104(x1, x2, x3) 42.39/12.88 42.39/12.88 a__plus(x1, x2) = a__plus(x1, x2) 42.39/12.88 42.39/12.88 a__U71(x1, x2) = a__U71(x1, x2) 42.39/12.88 42.39/12.88 a__U72(x1, x2) = a__U72(x1, x2) 42.39/12.88 42.39/12.88 a__U101(x1, x2, x3) = a__U101(x1, x2, x3) 42.39/12.88 42.39/12.88 a__U12(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 a__U13(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 a__U14(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 a__U15(x1, x2) = x1 42.39/12.88 42.39/12.88 a__U16(x1) = x1 42.39/12.88 42.39/12.88 a__U22(x1, x2) = x1 42.39/12.88 42.39/12.88 a__U23(x1) = x1 42.39/12.88 42.39/12.88 a__U32(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 a__U33(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 a__U34(x1, x2, x3) = x1 42.39/12.88 42.39/12.88 a__U35(x1, x2) = x1 42.39/12.88 42.39/12.88 a__U36(x1) = x1 42.39/12.88 42.39/12.88 a__U42(x1) = x1 42.39/12.88 42.39/12.88 a__U62(x1) = x1 42.39/12.88 42.39/12.88 a__U81(x1, x2, x3) = a__U81(x1, x2, x3) 42.39/12.88 42.39/12.88 a__U82(x1, x2, x3) = a__U82(x1, x2, x3) 42.39/12.88 42.39/12.88 a__U83(x1, x2, x3) = a__U83(x1, x2, x3) 42.39/12.88 42.39/12.88 a__U84(x1, x2, x3) = a__U84(x1, x2, x3) 42.39/12.88 42.39/12.88 a__U91(x1, x2) = a__U91(x1, x2) 42.39/12.88 42.39/12.88 a__U92(x1) = x1 42.39/12.88 42.39/12.88 42.39/12.88 Recursive path order with status [RPO]. 42.39/12.88 Quasi-Precedence: [A__U102_3, A__U103_3, A__U104_3, a__x_2, U101_3, A__U101_3, U102_3, U103_3, U104_3, A__X_2, x_2, a__U102_3, a__U103_3, a__U104_3, a__U101_3] > [plus_2, U81_3, U82_3, U83_3, U84_3, a__plus_2, a__U81_3, a__U82_3, a__U83_3, a__U84_3] > [A__PLUS_2, A__U81_3, A__U82_3, A__U83_3, A__U84_2] > [tt, a__isNat, a__isNatKind, U91_2, isNat, isNatKind, a__U91_2] > 0 > [U71_2, U72_2, a__U71_2, a__U72_2] 42.39/12.88 [A__U102_3, A__U103_3, A__U104_3, a__x_2, U101_3, A__U101_3, U102_3, U103_3, U104_3, A__X_2, x_2, a__U102_3, a__U103_3, a__U104_3, a__U101_3] > [plus_2, U81_3, U82_3, U83_3, U84_3, a__plus_2, a__U81_3, a__U82_3, a__U83_3, a__U84_3] > s_1 > [tt, a__isNat, a__isNatKind, U91_2, isNat, isNatKind, a__U91_2] > 0 > [U71_2, U72_2, a__U71_2, a__U72_2] 42.39/12.88 42.39/12.88 Status: A__U102_3: multiset status 42.39/12.88 tt: multiset status 42.39/12.88 A__U103_3: multiset status 42.39/12.88 a__isNat: multiset status 42.39/12.88 A__U104_3: multiset status 42.39/12.88 a__isNatKind: multiset status 42.39/12.88 A__PLUS_2: multiset status 42.39/12.88 a__x_2: multiset status 42.39/12.88 0: multiset status 42.39/12.88 U101_3: multiset status 42.39/12.88 A__U101_3: multiset status 42.39/12.88 U102_3: multiset status 42.39/12.88 U103_3: multiset status 42.39/12.88 U104_3: multiset status 42.39/12.88 A__X_2: multiset status 42.39/12.88 s_1: [1] 42.39/12.88 plus_2: multiset status 42.39/12.88 A__U81_3: multiset status 42.39/12.88 A__U82_3: multiset status 42.39/12.88 A__U83_3: multiset status 42.39/12.88 A__U84_2: multiset status 42.39/12.88 x_2: multiset status 42.39/12.88 U71_2: multiset status 42.39/12.88 U72_2: multiset status 42.39/12.88 U81_3: multiset status 42.39/12.88 U82_3: multiset status 42.39/12.88 U83_3: multiset status 42.39/12.88 U84_3: multiset status 42.39/12.88 U91_2: multiset status 42.39/12.88 isNat: multiset status 42.39/12.88 isNatKind: multiset status 42.39/12.88 a__U102_3: multiset status 42.39/12.88 a__U103_3: multiset status 42.39/12.88 a__U104_3: multiset status 42.39/12.88 a__plus_2: multiset status 42.39/12.88 a__U71_2: multiset status 42.39/12.88 a__U72_2: multiset status 42.39/12.88 a__U101_3: multiset status 42.39/12.88 a__U81_3: multiset status 42.39/12.88 a__U82_3: multiset status 42.39/12.88 a__U83_3: multiset status 42.39/12.88 a__U84_3: multiset status 42.39/12.88 a__U91_2: multiset status 42.39/12.88 42.39/12.88 42.39/12.88 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.39/12.88 42.39/12.88 a__isNat(0) -> tt 42.39/12.88 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.88 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.88 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.88 a__isNat(X) -> isNat(X) 42.39/12.88 a__isNatKind(0) -> tt 42.39/12.88 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.88 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.88 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.88 a__isNatKind(X) -> isNatKind(X) 42.39/12.88 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.88 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.88 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.88 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.88 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.88 a__U72(tt, N) -> mark(N) 42.39/12.88 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.88 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.88 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.88 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.88 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.88 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.88 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.88 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.88 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.88 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.88 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.88 mark(isNat(X)) -> a__isNat(X) 42.39/12.88 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.88 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.88 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.88 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.88 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.88 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.88 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.88 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.88 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.88 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.88 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.88 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.88 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.88 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.88 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.88 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.88 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.88 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.88 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.88 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.88 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.88 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.88 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.88 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.88 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.88 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.88 mark(tt) -> tt 42.39/12.88 mark(s(X)) -> s(mark(X)) 42.39/12.88 mark(0) -> 0 42.39/12.88 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.88 a__x(X1, X2) -> x(X1, X2) 42.39/12.88 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.88 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.88 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.88 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.88 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.88 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.88 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.88 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.88 a__U16(tt) -> tt 42.39/12.88 a__U16(X) -> U16(X) 42.39/12.88 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.88 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.88 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.88 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.88 a__U23(tt) -> tt 42.39/12.88 a__U23(X) -> U23(X) 42.39/12.88 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.88 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.88 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.88 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.88 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.88 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.88 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.88 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.88 a__U36(tt) -> tt 42.39/12.88 a__U36(X) -> U36(X) 42.39/12.88 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.88 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.88 a__U42(tt) -> tt 42.39/12.88 a__U42(X) -> U42(X) 42.39/12.88 a__U51(tt) -> tt 42.39/12.88 a__U51(X) -> U51(X) 42.39/12.88 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.88 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.88 a__U62(tt) -> tt 42.39/12.88 a__U62(X) -> U62(X) 42.39/12.88 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.88 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.88 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.88 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.88 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.88 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.88 a__U92(tt) -> 0 42.39/12.88 a__U92(X) -> U92(X) 42.39/12.88 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.88 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.88 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.88 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.88 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.88 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.88 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.88 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.88 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.88 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.88 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.88 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.88 42.39/12.88 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (21) 42.39/12.88 Obligation: 42.39/12.88 Q DP problem: 42.39/12.88 The TRS P consists of the following rules: 42.39/12.88 42.39/12.88 A__U102(tt, M, N) -> A__U103(a__isNat(N), M, N) 42.39/12.88 A__U103(tt, M, N) -> A__U104(a__isNatKind(N), M, N) 42.39/12.88 A__U71(tt, N) -> A__U72(a__isNatKind(N), N) 42.39/12.88 A__U72(tt, N) -> MARK(N) 42.39/12.88 MARK(U101(X1, X2, X3)) -> A__U101(mark(X1), X2, X3) 42.39/12.88 A__U101(tt, M, N) -> A__U102(a__isNatKind(M), M, N) 42.39/12.88 MARK(U102(X1, X2, X3)) -> A__U102(mark(X1), X2, X3) 42.39/12.88 MARK(U103(X1, X2, X3)) -> A__U103(mark(X1), X2, X3) 42.39/12.88 MARK(U104(X1, X2, X3)) -> A__U104(mark(X1), X2, X3) 42.39/12.88 A__U81(tt, M, N) -> A__U82(a__isNatKind(M), M, N) 42.39/12.88 A__U82(tt, M, N) -> A__U83(a__isNat(N), M, N) 42.39/12.88 A__U84(tt, M, N) -> A__PLUS(mark(N), mark(M)) 42.39/12.88 MARK(x(X1, X2)) -> A__X(mark(X1), mark(X2)) 42.39/12.88 MARK(U11(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U12(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U13(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U14(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U15(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U16(X)) -> MARK(X) 42.39/12.88 MARK(U21(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U22(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U23(X)) -> MARK(X) 42.39/12.88 MARK(U31(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U32(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U33(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U34(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U35(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U36(X)) -> MARK(X) 42.39/12.88 MARK(U41(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U42(X)) -> MARK(X) 42.39/12.88 MARK(U51(X)) -> MARK(X) 42.39/12.88 MARK(U61(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U62(X)) -> MARK(X) 42.39/12.88 MARK(U92(X)) -> MARK(X) 42.39/12.88 42.39/12.88 The TRS R consists of the following rules: 42.39/12.88 42.39/12.88 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.88 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.88 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.88 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.88 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.88 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.88 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.88 a__U16(tt) -> tt 42.39/12.88 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.88 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.88 a__U23(tt) -> tt 42.39/12.88 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.88 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.88 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.88 a__U36(tt) -> tt 42.39/12.88 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.88 a__U42(tt) -> tt 42.39/12.88 a__U51(tt) -> tt 42.39/12.88 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.88 a__U62(tt) -> tt 42.39/12.88 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.88 a__U72(tt, N) -> mark(N) 42.39/12.88 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.88 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.88 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.88 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.88 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.88 a__U92(tt) -> 0 42.39/12.88 a__isNat(0) -> tt 42.39/12.88 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.88 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.88 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.88 a__isNatKind(0) -> tt 42.39/12.88 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.88 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.88 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.88 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.88 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.88 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.88 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.88 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.88 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.88 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.88 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.88 mark(isNat(X)) -> a__isNat(X) 42.39/12.88 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.88 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.88 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.88 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.88 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.88 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.88 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.88 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.88 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.88 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.88 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.88 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.88 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.88 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.88 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.88 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.88 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.88 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.88 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.88 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.88 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.88 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.88 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.88 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.88 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.88 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.88 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.88 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.88 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.88 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.88 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.88 mark(tt) -> tt 42.39/12.88 mark(s(X)) -> s(mark(X)) 42.39/12.88 mark(0) -> 0 42.39/12.88 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.88 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.88 a__isNatKind(X) -> isNatKind(X) 42.39/12.88 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.88 a__isNat(X) -> isNat(X) 42.39/12.88 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.88 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.88 a__x(X1, X2) -> x(X1, X2) 42.39/12.88 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.88 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.88 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.88 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.88 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.88 a__U16(X) -> U16(X) 42.39/12.88 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.88 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.88 a__U23(X) -> U23(X) 42.39/12.88 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.88 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.88 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.88 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.88 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.88 a__U36(X) -> U36(X) 42.39/12.88 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.88 a__U42(X) -> U42(X) 42.39/12.88 a__U51(X) -> U51(X) 42.39/12.88 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.88 a__U62(X) -> U62(X) 42.39/12.88 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.88 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.88 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.88 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.88 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.88 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.88 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.88 a__U92(X) -> U92(X) 42.39/12.88 42.39/12.88 The set Q consists of the following terms: 42.39/12.88 42.39/12.88 mark(U101(x0, x1, x2)) 42.39/12.88 mark(U102(x0, x1, x2)) 42.39/12.88 mark(isNatKind(x0)) 42.39/12.88 mark(U103(x0, x1, x2)) 42.39/12.88 mark(isNat(x0)) 42.39/12.88 mark(U104(x0, x1, x2)) 42.39/12.88 mark(plus(x0, x1)) 42.39/12.88 mark(x(x0, x1)) 42.39/12.88 mark(U11(x0, x1, x2)) 42.39/12.88 mark(U12(x0, x1, x2)) 42.39/12.88 mark(U13(x0, x1, x2)) 42.39/12.88 mark(U14(x0, x1, x2)) 42.39/12.88 mark(U15(x0, x1)) 42.39/12.88 mark(U16(x0)) 42.39/12.88 mark(U21(x0, x1)) 42.39/12.88 mark(U22(x0, x1)) 42.39/12.88 mark(U23(x0)) 42.39/12.88 mark(U31(x0, x1, x2)) 42.39/12.88 mark(U32(x0, x1, x2)) 42.39/12.88 mark(U33(x0, x1, x2)) 42.39/12.88 mark(U34(x0, x1, x2)) 42.39/12.88 mark(U35(x0, x1)) 42.39/12.88 mark(U36(x0)) 42.39/12.88 mark(U41(x0, x1)) 42.39/12.88 mark(U42(x0)) 42.39/12.88 mark(U51(x0)) 42.39/12.88 mark(U61(x0, x1)) 42.39/12.88 mark(U62(x0)) 42.39/12.88 mark(U71(x0, x1)) 42.39/12.88 mark(U72(x0, x1)) 42.39/12.88 mark(U81(x0, x1, x2)) 42.39/12.88 mark(U82(x0, x1, x2)) 42.39/12.88 mark(U83(x0, x1, x2)) 42.39/12.88 mark(U84(x0, x1, x2)) 42.39/12.88 mark(U91(x0, x1)) 42.39/12.88 mark(U92(x0)) 42.39/12.88 mark(tt) 42.39/12.88 mark(s(x0)) 42.39/12.88 mark(0) 42.39/12.88 a__U101(x0, x1, x2) 42.39/12.88 a__U102(x0, x1, x2) 42.39/12.88 a__isNatKind(x0) 42.39/12.88 a__U103(x0, x1, x2) 42.39/12.88 a__isNat(x0) 42.39/12.88 a__U104(x0, x1, x2) 42.39/12.88 a__plus(x0, x1) 42.39/12.88 a__x(x0, x1) 42.39/12.88 a__U11(x0, x1, x2) 42.39/12.88 a__U12(x0, x1, x2) 42.39/12.88 a__U13(x0, x1, x2) 42.39/12.88 a__U14(x0, x1, x2) 42.39/12.88 a__U15(x0, x1) 42.39/12.88 a__U16(x0) 42.39/12.88 a__U21(x0, x1) 42.39/12.88 a__U22(x0, x1) 42.39/12.88 a__U23(x0) 42.39/12.88 a__U31(x0, x1, x2) 42.39/12.88 a__U32(x0, x1, x2) 42.39/12.88 a__U33(x0, x1, x2) 42.39/12.88 a__U34(x0, x1, x2) 42.39/12.88 a__U35(x0, x1) 42.39/12.88 a__U36(x0) 42.39/12.88 a__U41(x0, x1) 42.39/12.88 a__U42(x0) 42.39/12.88 a__U51(x0) 42.39/12.88 a__U61(x0, x1) 42.39/12.88 a__U62(x0) 42.39/12.88 a__U71(x0, x1) 42.39/12.88 a__U72(x0, x1) 42.39/12.88 a__U81(x0, x1, x2) 42.39/12.88 a__U82(x0, x1, x2) 42.39/12.88 a__U83(x0, x1, x2) 42.39/12.88 a__U84(x0, x1, x2) 42.39/12.88 a__U91(x0, x1) 42.39/12.88 a__U92(x0) 42.39/12.88 42.39/12.88 We have to consider all minimal (P,Q,R)-chains. 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (22) DependencyGraphProof (EQUIVALENT) 42.39/12.88 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 13 less nodes. 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (23) 42.39/12.88 Obligation: 42.39/12.88 Q DP problem: 42.39/12.88 The TRS P consists of the following rules: 42.39/12.88 42.39/12.88 MARK(U12(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U11(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U13(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U14(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U15(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U16(X)) -> MARK(X) 42.39/12.88 MARK(U21(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U22(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U23(X)) -> MARK(X) 42.39/12.88 MARK(U31(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U32(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U33(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U34(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U35(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U36(X)) -> MARK(X) 42.39/12.88 MARK(U41(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U42(X)) -> MARK(X) 42.39/12.88 MARK(U51(X)) -> MARK(X) 42.39/12.88 MARK(U61(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U62(X)) -> MARK(X) 42.39/12.88 MARK(U92(X)) -> MARK(X) 42.39/12.88 42.39/12.88 The TRS R consists of the following rules: 42.39/12.88 42.39/12.88 a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) 42.39/12.88 a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) 42.39/12.88 a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) 42.39/12.88 a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 42.39/12.88 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 42.39/12.88 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 42.39/12.88 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 42.39/12.88 a__U16(tt) -> tt 42.39/12.88 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.39/12.88 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.39/12.88 a__U23(tt) -> tt 42.39/12.88 a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) 42.39/12.88 a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) 42.39/12.88 a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) 42.39/12.88 a__U35(tt, V2) -> a__U36(a__isNat(V2)) 42.39/12.88 a__U36(tt) -> tt 42.39/12.88 a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) 42.39/12.88 a__U42(tt) -> tt 42.39/12.88 a__U51(tt) -> tt 42.39/12.88 a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) 42.39/12.88 a__U62(tt) -> tt 42.39/12.88 a__U71(tt, N) -> a__U72(a__isNatKind(N), N) 42.39/12.88 a__U72(tt, N) -> mark(N) 42.39/12.88 a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) 42.39/12.88 a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) 42.39/12.88 a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) 42.39/12.88 a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) 42.39/12.88 a__U91(tt, N) -> a__U92(a__isNatKind(N)) 42.39/12.88 a__U92(tt) -> 0 42.39/12.88 a__isNat(0) -> tt 42.39/12.88 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 42.39/12.88 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.39/12.88 a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) 42.39/12.88 a__isNatKind(0) -> tt 42.39/12.88 a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) 42.39/12.88 a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) 42.39/12.88 a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.39/12.88 a__plus(N, 0) -> a__U71(a__isNat(N), N) 42.39/12.88 a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) 42.39/12.88 a__x(N, 0) -> a__U91(a__isNat(N), N) 42.39/12.88 a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) 42.39/12.88 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.39/12.88 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.39/12.88 mark(isNatKind(X)) -> a__isNatKind(X) 42.39/12.88 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.39/12.88 mark(isNat(X)) -> a__isNat(X) 42.39/12.88 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.39/12.88 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 42.39/12.88 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 42.39/12.88 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 42.39/12.88 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 42.39/12.88 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 42.39/12.88 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 42.39/12.88 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 42.39/12.88 mark(U16(X)) -> a__U16(mark(X)) 42.39/12.88 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.39/12.88 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.39/12.88 mark(U23(X)) -> a__U23(mark(X)) 42.39/12.88 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 42.39/12.88 mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) 42.39/12.88 mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) 42.39/12.88 mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) 42.39/12.88 mark(U35(X1, X2)) -> a__U35(mark(X1), X2) 42.39/12.88 mark(U36(X)) -> a__U36(mark(X)) 42.39/12.88 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 42.39/12.88 mark(U42(X)) -> a__U42(mark(X)) 42.39/12.88 mark(U51(X)) -> a__U51(mark(X)) 42.39/12.88 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.39/12.88 mark(U62(X)) -> a__U62(mark(X)) 42.39/12.88 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 42.39/12.88 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 42.39/12.88 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 42.39/12.88 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 42.39/12.88 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 42.39/12.88 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 42.39/12.88 mark(U91(X1, X2)) -> a__U91(mark(X1), X2) 42.39/12.88 mark(U92(X)) -> a__U92(mark(X)) 42.39/12.88 mark(tt) -> tt 42.39/12.88 mark(s(X)) -> s(mark(X)) 42.39/12.88 mark(0) -> 0 42.39/12.88 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.39/12.88 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.39/12.88 a__isNatKind(X) -> isNatKind(X) 42.39/12.88 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.39/12.88 a__isNat(X) -> isNat(X) 42.39/12.88 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.39/12.88 a__plus(X1, X2) -> plus(X1, X2) 42.39/12.88 a__x(X1, X2) -> x(X1, X2) 42.39/12.88 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 42.39/12.88 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 42.39/12.88 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 42.39/12.88 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 42.39/12.88 a__U15(X1, X2) -> U15(X1, X2) 42.39/12.88 a__U16(X) -> U16(X) 42.39/12.88 a__U21(X1, X2) -> U21(X1, X2) 42.39/12.88 a__U22(X1, X2) -> U22(X1, X2) 42.39/12.88 a__U23(X) -> U23(X) 42.39/12.88 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 42.39/12.88 a__U32(X1, X2, X3) -> U32(X1, X2, X3) 42.39/12.88 a__U33(X1, X2, X3) -> U33(X1, X2, X3) 42.39/12.88 a__U34(X1, X2, X3) -> U34(X1, X2, X3) 42.39/12.88 a__U35(X1, X2) -> U35(X1, X2) 42.39/12.88 a__U36(X) -> U36(X) 42.39/12.88 a__U41(X1, X2) -> U41(X1, X2) 42.39/12.88 a__U42(X) -> U42(X) 42.39/12.88 a__U51(X) -> U51(X) 42.39/12.88 a__U61(X1, X2) -> U61(X1, X2) 42.39/12.88 a__U62(X) -> U62(X) 42.39/12.88 a__U71(X1, X2) -> U71(X1, X2) 42.39/12.88 a__U72(X1, X2) -> U72(X1, X2) 42.39/12.88 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 42.39/12.88 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 42.39/12.88 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 42.39/12.88 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 42.39/12.88 a__U91(X1, X2) -> U91(X1, X2) 42.39/12.88 a__U92(X) -> U92(X) 42.39/12.88 42.39/12.88 The set Q consists of the following terms: 42.39/12.88 42.39/12.88 mark(U101(x0, x1, x2)) 42.39/12.88 mark(U102(x0, x1, x2)) 42.39/12.88 mark(isNatKind(x0)) 42.39/12.88 mark(U103(x0, x1, x2)) 42.39/12.88 mark(isNat(x0)) 42.39/12.88 mark(U104(x0, x1, x2)) 42.39/12.88 mark(plus(x0, x1)) 42.39/12.88 mark(x(x0, x1)) 42.39/12.88 mark(U11(x0, x1, x2)) 42.39/12.88 mark(U12(x0, x1, x2)) 42.39/12.88 mark(U13(x0, x1, x2)) 42.39/12.88 mark(U14(x0, x1, x2)) 42.39/12.88 mark(U15(x0, x1)) 42.39/12.88 mark(U16(x0)) 42.39/12.88 mark(U21(x0, x1)) 42.39/12.88 mark(U22(x0, x1)) 42.39/12.88 mark(U23(x0)) 42.39/12.88 mark(U31(x0, x1, x2)) 42.39/12.88 mark(U32(x0, x1, x2)) 42.39/12.88 mark(U33(x0, x1, x2)) 42.39/12.88 mark(U34(x0, x1, x2)) 42.39/12.88 mark(U35(x0, x1)) 42.39/12.88 mark(U36(x0)) 42.39/12.88 mark(U41(x0, x1)) 42.39/12.88 mark(U42(x0)) 42.39/12.88 mark(U51(x0)) 42.39/12.88 mark(U61(x0, x1)) 42.39/12.88 mark(U62(x0)) 42.39/12.88 mark(U71(x0, x1)) 42.39/12.88 mark(U72(x0, x1)) 42.39/12.88 mark(U81(x0, x1, x2)) 42.39/12.88 mark(U82(x0, x1, x2)) 42.39/12.88 mark(U83(x0, x1, x2)) 42.39/12.88 mark(U84(x0, x1, x2)) 42.39/12.88 mark(U91(x0, x1)) 42.39/12.88 mark(U92(x0)) 42.39/12.88 mark(tt) 42.39/12.88 mark(s(x0)) 42.39/12.88 mark(0) 42.39/12.88 a__U101(x0, x1, x2) 42.39/12.88 a__U102(x0, x1, x2) 42.39/12.88 a__isNatKind(x0) 42.39/12.88 a__U103(x0, x1, x2) 42.39/12.88 a__isNat(x0) 42.39/12.88 a__U104(x0, x1, x2) 42.39/12.88 a__plus(x0, x1) 42.39/12.88 a__x(x0, x1) 42.39/12.88 a__U11(x0, x1, x2) 42.39/12.88 a__U12(x0, x1, x2) 42.39/12.88 a__U13(x0, x1, x2) 42.39/12.88 a__U14(x0, x1, x2) 42.39/12.88 a__U15(x0, x1) 42.39/12.88 a__U16(x0) 42.39/12.88 a__U21(x0, x1) 42.39/12.88 a__U22(x0, x1) 42.39/12.88 a__U23(x0) 42.39/12.88 a__U31(x0, x1, x2) 42.39/12.88 a__U32(x0, x1, x2) 42.39/12.88 a__U33(x0, x1, x2) 42.39/12.88 a__U34(x0, x1, x2) 42.39/12.88 a__U35(x0, x1) 42.39/12.88 a__U36(x0) 42.39/12.88 a__U41(x0, x1) 42.39/12.88 a__U42(x0) 42.39/12.88 a__U51(x0) 42.39/12.88 a__U61(x0, x1) 42.39/12.88 a__U62(x0) 42.39/12.88 a__U71(x0, x1) 42.39/12.88 a__U72(x0, x1) 42.39/12.88 a__U81(x0, x1, x2) 42.39/12.88 a__U82(x0, x1, x2) 42.39/12.88 a__U83(x0, x1, x2) 42.39/12.88 a__U84(x0, x1, x2) 42.39/12.88 a__U91(x0, x1) 42.39/12.88 a__U92(x0) 42.39/12.88 42.39/12.88 We have to consider all minimal (P,Q,R)-chains. 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (24) UsableRulesProof (EQUIVALENT) 42.39/12.88 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. 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (25) 42.39/12.88 Obligation: 42.39/12.88 Q DP problem: 42.39/12.88 The TRS P consists of the following rules: 42.39/12.88 42.39/12.88 MARK(U12(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U11(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U13(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U14(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U15(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U16(X)) -> MARK(X) 42.39/12.88 MARK(U21(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U22(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U23(X)) -> MARK(X) 42.39/12.88 MARK(U31(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U32(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U33(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U34(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U35(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U36(X)) -> MARK(X) 42.39/12.88 MARK(U41(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U42(X)) -> MARK(X) 42.39/12.88 MARK(U51(X)) -> MARK(X) 42.39/12.88 MARK(U61(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U62(X)) -> MARK(X) 42.39/12.88 MARK(U92(X)) -> MARK(X) 42.39/12.88 42.39/12.88 R is empty. 42.39/12.88 The set Q consists of the following terms: 42.39/12.88 42.39/12.88 mark(U101(x0, x1, x2)) 42.39/12.88 mark(U102(x0, x1, x2)) 42.39/12.88 mark(isNatKind(x0)) 42.39/12.88 mark(U103(x0, x1, x2)) 42.39/12.88 mark(isNat(x0)) 42.39/12.88 mark(U104(x0, x1, x2)) 42.39/12.88 mark(plus(x0, x1)) 42.39/12.88 mark(x(x0, x1)) 42.39/12.88 mark(U11(x0, x1, x2)) 42.39/12.88 mark(U12(x0, x1, x2)) 42.39/12.88 mark(U13(x0, x1, x2)) 42.39/12.88 mark(U14(x0, x1, x2)) 42.39/12.88 mark(U15(x0, x1)) 42.39/12.88 mark(U16(x0)) 42.39/12.88 mark(U21(x0, x1)) 42.39/12.88 mark(U22(x0, x1)) 42.39/12.88 mark(U23(x0)) 42.39/12.88 mark(U31(x0, x1, x2)) 42.39/12.88 mark(U32(x0, x1, x2)) 42.39/12.88 mark(U33(x0, x1, x2)) 42.39/12.88 mark(U34(x0, x1, x2)) 42.39/12.88 mark(U35(x0, x1)) 42.39/12.88 mark(U36(x0)) 42.39/12.88 mark(U41(x0, x1)) 42.39/12.88 mark(U42(x0)) 42.39/12.88 mark(U51(x0)) 42.39/12.88 mark(U61(x0, x1)) 42.39/12.88 mark(U62(x0)) 42.39/12.88 mark(U71(x0, x1)) 42.39/12.88 mark(U72(x0, x1)) 42.39/12.88 mark(U81(x0, x1, x2)) 42.39/12.88 mark(U82(x0, x1, x2)) 42.39/12.88 mark(U83(x0, x1, x2)) 42.39/12.88 mark(U84(x0, x1, x2)) 42.39/12.88 mark(U91(x0, x1)) 42.39/12.88 mark(U92(x0)) 42.39/12.88 mark(tt) 42.39/12.88 mark(s(x0)) 42.39/12.88 mark(0) 42.39/12.88 a__U101(x0, x1, x2) 42.39/12.88 a__U102(x0, x1, x2) 42.39/12.88 a__isNatKind(x0) 42.39/12.88 a__U103(x0, x1, x2) 42.39/12.88 a__isNat(x0) 42.39/12.88 a__U104(x0, x1, x2) 42.39/12.88 a__plus(x0, x1) 42.39/12.88 a__x(x0, x1) 42.39/12.88 a__U11(x0, x1, x2) 42.39/12.88 a__U12(x0, x1, x2) 42.39/12.88 a__U13(x0, x1, x2) 42.39/12.88 a__U14(x0, x1, x2) 42.39/12.88 a__U15(x0, x1) 42.39/12.88 a__U16(x0) 42.39/12.88 a__U21(x0, x1) 42.39/12.88 a__U22(x0, x1) 42.39/12.88 a__U23(x0) 42.39/12.88 a__U31(x0, x1, x2) 42.39/12.88 a__U32(x0, x1, x2) 42.39/12.88 a__U33(x0, x1, x2) 42.39/12.88 a__U34(x0, x1, x2) 42.39/12.88 a__U35(x0, x1) 42.39/12.88 a__U36(x0) 42.39/12.88 a__U41(x0, x1) 42.39/12.88 a__U42(x0) 42.39/12.88 a__U51(x0) 42.39/12.88 a__U61(x0, x1) 42.39/12.88 a__U62(x0) 42.39/12.88 a__U71(x0, x1) 42.39/12.88 a__U72(x0, x1) 42.39/12.88 a__U81(x0, x1, x2) 42.39/12.88 a__U82(x0, x1, x2) 42.39/12.88 a__U83(x0, x1, x2) 42.39/12.88 a__U84(x0, x1, x2) 42.39/12.88 a__U91(x0, x1) 42.39/12.88 a__U92(x0) 42.39/12.88 42.39/12.88 We have to consider all minimal (P,Q,R)-chains. 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (26) QReductionProof (EQUIVALENT) 42.39/12.88 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 42.39/12.88 42.39/12.88 mark(U101(x0, x1, x2)) 42.39/12.88 mark(U102(x0, x1, x2)) 42.39/12.88 mark(isNatKind(x0)) 42.39/12.88 mark(U103(x0, x1, x2)) 42.39/12.88 mark(isNat(x0)) 42.39/12.88 mark(U104(x0, x1, x2)) 42.39/12.88 mark(plus(x0, x1)) 42.39/12.88 mark(x(x0, x1)) 42.39/12.88 mark(U11(x0, x1, x2)) 42.39/12.88 mark(U12(x0, x1, x2)) 42.39/12.88 mark(U13(x0, x1, x2)) 42.39/12.88 mark(U14(x0, x1, x2)) 42.39/12.88 mark(U15(x0, x1)) 42.39/12.88 mark(U16(x0)) 42.39/12.88 mark(U21(x0, x1)) 42.39/12.88 mark(U22(x0, x1)) 42.39/12.88 mark(U23(x0)) 42.39/12.88 mark(U31(x0, x1, x2)) 42.39/12.88 mark(U32(x0, x1, x2)) 42.39/12.88 mark(U33(x0, x1, x2)) 42.39/12.88 mark(U34(x0, x1, x2)) 42.39/12.88 mark(U35(x0, x1)) 42.39/12.88 mark(U36(x0)) 42.39/12.88 mark(U41(x0, x1)) 42.39/12.88 mark(U42(x0)) 42.39/12.88 mark(U51(x0)) 42.39/12.88 mark(U61(x0, x1)) 42.39/12.88 mark(U62(x0)) 42.39/12.88 mark(U71(x0, x1)) 42.39/12.88 mark(U72(x0, x1)) 42.39/12.88 mark(U81(x0, x1, x2)) 42.39/12.88 mark(U82(x0, x1, x2)) 42.39/12.88 mark(U83(x0, x1, x2)) 42.39/12.88 mark(U84(x0, x1, x2)) 42.39/12.88 mark(U91(x0, x1)) 42.39/12.88 mark(U92(x0)) 42.39/12.88 mark(tt) 42.39/12.88 mark(s(x0)) 42.39/12.88 mark(0) 42.39/12.88 a__U101(x0, x1, x2) 42.39/12.88 a__U102(x0, x1, x2) 42.39/12.88 a__isNatKind(x0) 42.39/12.88 a__U103(x0, x1, x2) 42.39/12.88 a__isNat(x0) 42.39/12.88 a__U104(x0, x1, x2) 42.39/12.88 a__plus(x0, x1) 42.39/12.88 a__x(x0, x1) 42.39/12.88 a__U11(x0, x1, x2) 42.39/12.88 a__U12(x0, x1, x2) 42.39/12.88 a__U13(x0, x1, x2) 42.39/12.88 a__U14(x0, x1, x2) 42.39/12.88 a__U15(x0, x1) 42.39/12.88 a__U16(x0) 42.39/12.88 a__U21(x0, x1) 42.39/12.88 a__U22(x0, x1) 42.39/12.88 a__U23(x0) 42.39/12.88 a__U31(x0, x1, x2) 42.39/12.88 a__U32(x0, x1, x2) 42.39/12.88 a__U33(x0, x1, x2) 42.39/12.88 a__U34(x0, x1, x2) 42.39/12.88 a__U35(x0, x1) 42.39/12.88 a__U36(x0) 42.39/12.88 a__U41(x0, x1) 42.39/12.88 a__U42(x0) 42.39/12.88 a__U51(x0) 42.39/12.88 a__U61(x0, x1) 42.39/12.88 a__U62(x0) 42.39/12.88 a__U71(x0, x1) 42.39/12.88 a__U72(x0, x1) 42.39/12.88 a__U81(x0, x1, x2) 42.39/12.88 a__U82(x0, x1, x2) 42.39/12.88 a__U83(x0, x1, x2) 42.39/12.88 a__U84(x0, x1, x2) 42.39/12.88 a__U91(x0, x1) 42.39/12.88 a__U92(x0) 42.39/12.88 42.39/12.88 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (27) 42.39/12.88 Obligation: 42.39/12.88 Q DP problem: 42.39/12.88 The TRS P consists of the following rules: 42.39/12.88 42.39/12.88 MARK(U12(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U11(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U13(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U14(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U15(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U16(X)) -> MARK(X) 42.39/12.88 MARK(U21(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U22(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U23(X)) -> MARK(X) 42.39/12.88 MARK(U31(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U32(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U33(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U34(X1, X2, X3)) -> MARK(X1) 42.39/12.88 MARK(U35(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U36(X)) -> MARK(X) 42.39/12.88 MARK(U41(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U42(X)) -> MARK(X) 42.39/12.88 MARK(U51(X)) -> MARK(X) 42.39/12.88 MARK(U61(X1, X2)) -> MARK(X1) 42.39/12.88 MARK(U62(X)) -> MARK(X) 42.39/12.88 MARK(U92(X)) -> MARK(X) 42.39/12.88 42.39/12.88 R is empty. 42.39/12.88 Q is empty. 42.39/12.88 We have to consider all minimal (P,Q,R)-chains. 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (28) QDPSizeChangeProof (EQUIVALENT) 42.39/12.88 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. 42.39/12.88 42.39/12.88 From the DPs we obtained the following set of size-change graphs: 42.39/12.88 *MARK(U12(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U11(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U13(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U14(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U15(X1, X2)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U16(X)) -> MARK(X) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U21(X1, X2)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U22(X1, X2)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U23(X)) -> MARK(X) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U31(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U32(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U33(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U34(X1, X2, X3)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U35(X1, X2)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U36(X)) -> MARK(X) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U41(X1, X2)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U42(X)) -> MARK(X) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U51(X)) -> MARK(X) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U61(X1, X2)) -> MARK(X1) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U62(X)) -> MARK(X) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 *MARK(U92(X)) -> MARK(X) 42.39/12.88 The graph contains the following edges 1 > 1 42.39/12.88 42.39/12.88 42.39/12.88 ---------------------------------------- 42.39/12.88 42.39/12.88 (29) 42.39/12.88 YES 42.53/12.94 EOF