WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 2822 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (6) BEST (7) proven lower bound (8) LowerBoundPropagationProof [FINISHED, 0 ms] (9) BOUNDS(n^1, INF) (10) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U11(tt, z0, z1) -> c(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0, z1) -> c2(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c3 A__U13(tt, z0, z1) -> c4(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c5 A__U14(tt, z0, z1) -> c6(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c7 A__U15(tt, z0) -> c8(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c9 A__U16(tt) -> c10 A__U16(z0) -> c11 A__U21(tt, z0) -> c12(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c13 A__U22(tt, z0) -> c14(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c15 A__U23(tt) -> c16 A__U23(z0) -> c17 A__U31(tt, z0) -> c18(A__U32(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U31(z0, z1) -> c19 A__U32(tt) -> c20 A__U32(z0) -> c21 A__U41(tt) -> c22 A__U41(z0) -> c23 A__U51(tt, z0) -> c24(A__U52(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U51(z0, z1) -> c25 A__U52(tt, z0) -> c26(MARK(z0)) A__U52(z0, z1) -> c27 A__U61(tt, z0, z1) -> c28(A__U62(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U61(z0, z1, z2) -> c29 A__U62(tt, z0, z1) -> c30(A__U63(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U62(z0, z1, z2) -> c31 A__U63(tt, z0, z1) -> c32(A__U64(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U63(z0, z1, z2) -> c33 A__U64(tt, z0, z1) -> c34(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U64(tt, z0, z1) -> c35(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U64(z0, z1, z2) -> c36 A__ISNAT(0) -> c37 A__ISNAT(plus(z0, z1)) -> c38(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c39(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c40 A__ISNATKIND(0) -> c41 A__ISNATKIND(plus(z0, z1)) -> c42(A__U31(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c43(A__U41(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c44 A__PLUS(z0, 0) -> c45(A__U51(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c46(A__U61(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c47 MARK(U11(z0, z1, z2)) -> c48(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c49(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c50(A__ISNATKIND(z0)) MARK(U13(z0, z1, z2)) -> c51(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c52(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c53(A__U15(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c54(A__ISNAT(z0)) MARK(U16(z0)) -> c55(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c56(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c57(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c58(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c59(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c60(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0)) -> c61(A__U41(mark(z0)), MARK(z0)) MARK(U51(z0, z1)) -> c62(A__U51(mark(z0), z1), MARK(z0)) MARK(U52(z0, z1)) -> c63(A__U52(mark(z0), z1), MARK(z0)) MARK(U61(z0, z1, z2)) -> c64(A__U61(mark(z0), z1, z2), MARK(z0)) MARK(U62(z0, z1, z2)) -> c65(A__U62(mark(z0), z1, z2), MARK(z0)) MARK(U63(z0, z1, z2)) -> c66(A__U63(mark(z0), z1, z2), MARK(z0)) MARK(U64(z0, z1, z2)) -> c67(A__U64(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c68(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c69(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c70 MARK(s(z0)) -> c71(MARK(z0)) MARK(0) -> c72 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0) -> a__U32(a__isNatKind(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt) -> tt a__U41(z0) -> U41(z0) a__U51(tt, z0) -> a__U52(a__isNatKind(z0), z0) a__U51(z0, z1) -> U51(z0, z1) a__U52(tt, z0) -> mark(z0) a__U52(z0, z1) -> U52(z0, z1) a__U61(tt, z0, z1) -> a__U62(a__isNatKind(z0), z0, z1) a__U61(z0, z1, z2) -> U61(z0, z1, z2) a__U62(tt, z0, z1) -> a__U63(a__isNat(z1), z0, z1) a__U62(z0, z1, z2) -> U62(z0, z1, z2) a__U63(tt, z0, z1) -> a__U64(a__isNatKind(z1), z0, z1) a__U63(z0, z1, z2) -> U63(z0, z1, z2) a__U64(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U64(z0, z1, z2) -> U64(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U31(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U41(a__isNatKind(z0)) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U51(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U61(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0)) -> a__U41(mark(z0)) mark(U51(z0, z1)) -> a__U51(mark(z0), z1) mark(U52(z0, z1)) -> a__U52(mark(z0), z1) mark(U61(z0, z1, z2)) -> a__U61(mark(z0), z1, z2) mark(U62(z0, z1, z2)) -> a__U62(mark(z0), z1, z2) mark(U63(z0, z1, z2)) -> a__U63(mark(z0), z1, z2) mark(U64(z0, z1, z2)) -> a__U64(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U11(tt, z0, z1) -> c(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0, z1) -> c2(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c3 A__U13(tt, z0, z1) -> c4(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c5 A__U14(tt, z0, z1) -> c6(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c7 A__U15(tt, z0) -> c8(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c9 A__U16(tt) -> c10 A__U16(z0) -> c11 A__U21(tt, z0) -> c12(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c13 A__U22(tt, z0) -> c14(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c15 A__U23(tt) -> c16 A__U23(z0) -> c17 A__U31(tt, z0) -> c18(A__U32(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U31(z0, z1) -> c19 A__U32(tt) -> c20 A__U32(z0) -> c21 A__U41(tt) -> c22 A__U41(z0) -> c23 A__U51(tt, z0) -> c24(A__U52(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U51(z0, z1) -> c25 A__U52(tt, z0) -> c26(MARK(z0)) A__U52(z0, z1) -> c27 A__U61(tt, z0, z1) -> c28(A__U62(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U61(z0, z1, z2) -> c29 A__U62(tt, z0, z1) -> c30(A__U63(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U62(z0, z1, z2) -> c31 A__U63(tt, z0, z1) -> c32(A__U64(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U63(z0, z1, z2) -> c33 A__U64(tt, z0, z1) -> c34(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U64(tt, z0, z1) -> c35(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U64(z0, z1, z2) -> c36 A__ISNAT(0) -> c37 A__ISNAT(plus(z0, z1)) -> c38(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c39(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c40 A__ISNATKIND(0) -> c41 A__ISNATKIND(plus(z0, z1)) -> c42(A__U31(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c43(A__U41(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c44 A__PLUS(z0, 0) -> c45(A__U51(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c46(A__U61(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c47 MARK(U11(z0, z1, z2)) -> c48(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c49(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c50(A__ISNATKIND(z0)) MARK(U13(z0, z1, z2)) -> c51(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c52(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c53(A__U15(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c54(A__ISNAT(z0)) MARK(U16(z0)) -> c55(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c56(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c57(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c58(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c59(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c60(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0)) -> c61(A__U41(mark(z0)), MARK(z0)) MARK(U51(z0, z1)) -> c62(A__U51(mark(z0), z1), MARK(z0)) MARK(U52(z0, z1)) -> c63(A__U52(mark(z0), z1), MARK(z0)) MARK(U61(z0, z1, z2)) -> c64(A__U61(mark(z0), z1, z2), MARK(z0)) MARK(U62(z0, z1, z2)) -> c65(A__U62(mark(z0), z1, z2), MARK(z0)) MARK(U63(z0, z1, z2)) -> c66(A__U63(mark(z0), z1, z2), MARK(z0)) MARK(U64(z0, z1, z2)) -> c67(A__U64(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c68(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c69(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c70 MARK(s(z0)) -> c71(MARK(z0)) MARK(0) -> c72 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0) -> a__U32(a__isNatKind(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt) -> tt a__U41(z0) -> U41(z0) a__U51(tt, z0) -> a__U52(a__isNatKind(z0), z0) a__U51(z0, z1) -> U51(z0, z1) a__U52(tt, z0) -> mark(z0) a__U52(z0, z1) -> U52(z0, z1) a__U61(tt, z0, z1) -> a__U62(a__isNatKind(z0), z0, z1) a__U61(z0, z1, z2) -> U61(z0, z1, z2) a__U62(tt, z0, z1) -> a__U63(a__isNat(z1), z0, z1) a__U62(z0, z1, z2) -> U62(z0, z1, z2) a__U63(tt, z0, z1) -> a__U64(a__isNatKind(z1), z0, z1) a__U63(z0, z1, z2) -> U63(z0, z1, z2) a__U64(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U64(z0, z1, z2) -> U64(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U31(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U41(a__isNatKind(z0)) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U51(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U61(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0)) -> a__U41(mark(z0)) mark(U51(z0, z1)) -> a__U51(mark(z0), z1) mark(U52(z0, z1)) -> a__U52(mark(z0), z1) mark(U61(z0, z1, z2)) -> a__U61(mark(z0), z1, z2) mark(U62(z0, z1, z2)) -> a__U62(mark(z0), z1, z2) mark(U63(z0, z1, z2)) -> a__U63(mark(z0), z1, z2) mark(U64(z0, z1, z2)) -> a__U64(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (4) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U11(tt, z0, z1) -> c(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0, z1) -> c2(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c3 A__U13(tt, z0, z1) -> c4(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c5 A__U14(tt, z0, z1) -> c6(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c7 A__U15(tt, z0) -> c8(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c9 A__U16(tt) -> c10 A__U16(z0) -> c11 A__U21(tt, z0) -> c12(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c13 A__U22(tt, z0) -> c14(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c15 A__U23(tt) -> c16 A__U23(z0) -> c17 A__U31(tt, z0) -> c18(A__U32(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U31(z0, z1) -> c19 A__U32(tt) -> c20 A__U32(z0) -> c21 A__U41(tt) -> c22 A__U41(z0) -> c23 A__U51(tt, z0) -> c24(A__U52(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U51(z0, z1) -> c25 A__U52(tt, z0) -> c26(MARK(z0)) A__U52(z0, z1) -> c27 A__U61(tt, z0, z1) -> c28(A__U62(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U61(z0, z1, z2) -> c29 A__U62(tt, z0, z1) -> c30(A__U63(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U62(z0, z1, z2) -> c31 A__U63(tt, z0, z1) -> c32(A__U64(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U63(z0, z1, z2) -> c33 A__U64(tt, z0, z1) -> c34(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U64(tt, z0, z1) -> c35(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U64(z0, z1, z2) -> c36 A__ISNAT(0) -> c37 A__ISNAT(plus(z0, z1)) -> c38(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c39(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c40 A__ISNATKIND(0) -> c41 A__ISNATKIND(plus(z0, z1)) -> c42(A__U31(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c43(A__U41(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c44 A__PLUS(z0, 0) -> c45(A__U51(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c46(A__U61(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c47 MARK(U11(z0, z1, z2)) -> c48(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c49(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c50(A__ISNATKIND(z0)) MARK(U13(z0, z1, z2)) -> c51(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c52(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c53(A__U15(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c54(A__ISNAT(z0)) MARK(U16(z0)) -> c55(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c56(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c57(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c58(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c59(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c60(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0)) -> c61(A__U41(mark(z0)), MARK(z0)) MARK(U51(z0, z1)) -> c62(A__U51(mark(z0), z1), MARK(z0)) MARK(U52(z0, z1)) -> c63(A__U52(mark(z0), z1), MARK(z0)) MARK(U61(z0, z1, z2)) -> c64(A__U61(mark(z0), z1, z2), MARK(z0)) MARK(U62(z0, z1, z2)) -> c65(A__U62(mark(z0), z1, z2), MARK(z0)) MARK(U63(z0, z1, z2)) -> c66(A__U63(mark(z0), z1, z2), MARK(z0)) MARK(U64(z0, z1, z2)) -> c67(A__U64(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c68(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c69(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c70 MARK(s(z0)) -> c71(MARK(z0)) MARK(0) -> c72 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0) -> a__U32(a__isNatKind(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt) -> tt a__U41(z0) -> U41(z0) a__U51(tt, z0) -> a__U52(a__isNatKind(z0), z0) a__U51(z0, z1) -> U51(z0, z1) a__U52(tt, z0) -> mark(z0) a__U52(z0, z1) -> U52(z0, z1) a__U61(tt, z0, z1) -> a__U62(a__isNatKind(z0), z0, z1) a__U61(z0, z1, z2) -> U61(z0, z1, z2) a__U62(tt, z0, z1) -> a__U63(a__isNat(z1), z0, z1) a__U62(z0, z1, z2) -> U62(z0, z1, z2) a__U63(tt, z0, z1) -> a__U64(a__isNatKind(z1), z0, z1) a__U63(z0, z1, z2) -> U63(z0, z1, z2) a__U64(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U64(z0, z1, z2) -> U64(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U31(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U41(a__isNatKind(z0)) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U51(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U61(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0)) -> a__U41(mark(z0)) mark(U51(z0, z1)) -> a__U51(mark(z0), z1) mark(U52(z0, z1)) -> a__U52(mark(z0), z1) mark(U61(z0, z1, z2)) -> a__U61(mark(z0), z1, z2) mark(U62(z0, z1, z2)) -> a__U62(mark(z0), z1, z2) mark(U63(z0, z1, z2)) -> a__U63(mark(z0), z1, z2) mark(U64(z0, z1, z2)) -> a__U64(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (5) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence MARK(U63(z0, z1, z2)) ->^+ c66(A__U63(mark(z0), z1, z2), MARK(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / U63(z0, z1, z2)]. The result substitution is [ ]. ---------------------------------------- (6) Complex Obligation (BEST) ---------------------------------------- (7) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U11(tt, z0, z1) -> c(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0, z1) -> c2(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c3 A__U13(tt, z0, z1) -> c4(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c5 A__U14(tt, z0, z1) -> c6(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c7 A__U15(tt, z0) -> c8(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c9 A__U16(tt) -> c10 A__U16(z0) -> c11 A__U21(tt, z0) -> c12(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c13 A__U22(tt, z0) -> c14(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c15 A__U23(tt) -> c16 A__U23(z0) -> c17 A__U31(tt, z0) -> c18(A__U32(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U31(z0, z1) -> c19 A__U32(tt) -> c20 A__U32(z0) -> c21 A__U41(tt) -> c22 A__U41(z0) -> c23 A__U51(tt, z0) -> c24(A__U52(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U51(z0, z1) -> c25 A__U52(tt, z0) -> c26(MARK(z0)) A__U52(z0, z1) -> c27 A__U61(tt, z0, z1) -> c28(A__U62(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U61(z0, z1, z2) -> c29 A__U62(tt, z0, z1) -> c30(A__U63(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U62(z0, z1, z2) -> c31 A__U63(tt, z0, z1) -> c32(A__U64(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U63(z0, z1, z2) -> c33 A__U64(tt, z0, z1) -> c34(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U64(tt, z0, z1) -> c35(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U64(z0, z1, z2) -> c36 A__ISNAT(0) -> c37 A__ISNAT(plus(z0, z1)) -> c38(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c39(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c40 A__ISNATKIND(0) -> c41 A__ISNATKIND(plus(z0, z1)) -> c42(A__U31(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c43(A__U41(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c44 A__PLUS(z0, 0) -> c45(A__U51(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c46(A__U61(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c47 MARK(U11(z0, z1, z2)) -> c48(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c49(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c50(A__ISNATKIND(z0)) MARK(U13(z0, z1, z2)) -> c51(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c52(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c53(A__U15(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c54(A__ISNAT(z0)) MARK(U16(z0)) -> c55(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c56(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c57(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c58(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c59(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c60(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0)) -> c61(A__U41(mark(z0)), MARK(z0)) MARK(U51(z0, z1)) -> c62(A__U51(mark(z0), z1), MARK(z0)) MARK(U52(z0, z1)) -> c63(A__U52(mark(z0), z1), MARK(z0)) MARK(U61(z0, z1, z2)) -> c64(A__U61(mark(z0), z1, z2), MARK(z0)) MARK(U62(z0, z1, z2)) -> c65(A__U62(mark(z0), z1, z2), MARK(z0)) MARK(U63(z0, z1, z2)) -> c66(A__U63(mark(z0), z1, z2), MARK(z0)) MARK(U64(z0, z1, z2)) -> c67(A__U64(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c68(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c69(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c70 MARK(s(z0)) -> c71(MARK(z0)) MARK(0) -> c72 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0) -> a__U32(a__isNatKind(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt) -> tt a__U41(z0) -> U41(z0) a__U51(tt, z0) -> a__U52(a__isNatKind(z0), z0) a__U51(z0, z1) -> U51(z0, z1) a__U52(tt, z0) -> mark(z0) a__U52(z0, z1) -> U52(z0, z1) a__U61(tt, z0, z1) -> a__U62(a__isNatKind(z0), z0, z1) a__U61(z0, z1, z2) -> U61(z0, z1, z2) a__U62(tt, z0, z1) -> a__U63(a__isNat(z1), z0, z1) a__U62(z0, z1, z2) -> U62(z0, z1, z2) a__U63(tt, z0, z1) -> a__U64(a__isNatKind(z1), z0, z1) a__U63(z0, z1, z2) -> U63(z0, z1, z2) a__U64(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U64(z0, z1, z2) -> U64(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U31(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U41(a__isNatKind(z0)) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U51(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U61(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0)) -> a__U41(mark(z0)) mark(U51(z0, z1)) -> a__U51(mark(z0), z1) mark(U52(z0, z1)) -> a__U52(mark(z0), z1) mark(U61(z0, z1, z2)) -> a__U61(mark(z0), z1, z2) mark(U62(z0, z1, z2)) -> a__U62(mark(z0), z1, z2) mark(U63(z0, z1, z2)) -> a__U63(mark(z0), z1, z2) mark(U64(z0, z1, z2)) -> a__U64(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (8) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (9) BOUNDS(n^1, INF) ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U11(tt, z0, z1) -> c(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0, z1) -> c2(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c3 A__U13(tt, z0, z1) -> c4(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c5 A__U14(tt, z0, z1) -> c6(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c7 A__U15(tt, z0) -> c8(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c9 A__U16(tt) -> c10 A__U16(z0) -> c11 A__U21(tt, z0) -> c12(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c13 A__U22(tt, z0) -> c14(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c15 A__U23(tt) -> c16 A__U23(z0) -> c17 A__U31(tt, z0) -> c18(A__U32(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U31(z0, z1) -> c19 A__U32(tt) -> c20 A__U32(z0) -> c21 A__U41(tt) -> c22 A__U41(z0) -> c23 A__U51(tt, z0) -> c24(A__U52(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U51(z0, z1) -> c25 A__U52(tt, z0) -> c26(MARK(z0)) A__U52(z0, z1) -> c27 A__U61(tt, z0, z1) -> c28(A__U62(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U61(z0, z1, z2) -> c29 A__U62(tt, z0, z1) -> c30(A__U63(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U62(z0, z1, z2) -> c31 A__U63(tt, z0, z1) -> c32(A__U64(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U63(z0, z1, z2) -> c33 A__U64(tt, z0, z1) -> c34(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U64(tt, z0, z1) -> c35(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U64(z0, z1, z2) -> c36 A__ISNAT(0) -> c37 A__ISNAT(plus(z0, z1)) -> c38(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c39(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c40 A__ISNATKIND(0) -> c41 A__ISNATKIND(plus(z0, z1)) -> c42(A__U31(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c43(A__U41(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c44 A__PLUS(z0, 0) -> c45(A__U51(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c46(A__U61(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c47 MARK(U11(z0, z1, z2)) -> c48(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c49(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c50(A__ISNATKIND(z0)) MARK(U13(z0, z1, z2)) -> c51(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c52(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c53(A__U15(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c54(A__ISNAT(z0)) MARK(U16(z0)) -> c55(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c56(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c57(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c58(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c59(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c60(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0)) -> c61(A__U41(mark(z0)), MARK(z0)) MARK(U51(z0, z1)) -> c62(A__U51(mark(z0), z1), MARK(z0)) MARK(U52(z0, z1)) -> c63(A__U52(mark(z0), z1), MARK(z0)) MARK(U61(z0, z1, z2)) -> c64(A__U61(mark(z0), z1, z2), MARK(z0)) MARK(U62(z0, z1, z2)) -> c65(A__U62(mark(z0), z1, z2), MARK(z0)) MARK(U63(z0, z1, z2)) -> c66(A__U63(mark(z0), z1, z2), MARK(z0)) MARK(U64(z0, z1, z2)) -> c67(A__U64(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c68(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c69(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c70 MARK(s(z0)) -> c71(MARK(z0)) MARK(0) -> c72 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0) -> a__U32(a__isNatKind(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt) -> tt a__U41(z0) -> U41(z0) a__U51(tt, z0) -> a__U52(a__isNatKind(z0), z0) a__U51(z0, z1) -> U51(z0, z1) a__U52(tt, z0) -> mark(z0) a__U52(z0, z1) -> U52(z0, z1) a__U61(tt, z0, z1) -> a__U62(a__isNatKind(z0), z0, z1) a__U61(z0, z1, z2) -> U61(z0, z1, z2) a__U62(tt, z0, z1) -> a__U63(a__isNat(z1), z0, z1) a__U62(z0, z1, z2) -> U62(z0, z1, z2) a__U63(tt, z0, z1) -> a__U64(a__isNatKind(z1), z0, z1) a__U63(z0, z1, z2) -> U63(z0, z1, z2) a__U64(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U64(z0, z1, z2) -> U64(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U31(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U41(a__isNatKind(z0)) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U51(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U61(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0)) -> a__U41(mark(z0)) mark(U51(z0, z1)) -> a__U51(mark(z0), z1) mark(U52(z0, z1)) -> a__U52(mark(z0), z1) mark(U61(z0, z1, z2)) -> a__U61(mark(z0), z1, z2) mark(U62(z0, z1, z2)) -> a__U62(mark(z0), z1, z2) mark(U63(z0, z1, z2)) -> a__U63(mark(z0), z1, z2) mark(U64(z0, z1, z2)) -> a__U64(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST