17.45/5.25 WORST_CASE(NON_POLY, ?) 17.45/5.27 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 17.45/5.27 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 17.45/5.27 17.45/5.27 17.45/5.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.45/5.27 17.45/5.27 (0) CpxTRS 17.45/5.27 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 17.45/5.27 (2) TRS for Loop Detection 17.45/5.27 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 17.45/5.27 (4) BEST 17.45/5.27 (5) proven lower bound 17.45/5.27 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 17.45/5.27 (7) BOUNDS(n^1, INF) 17.45/5.27 (8) TRS for Loop Detection 17.45/5.27 (9) DecreasingLoopProof [FINISHED, 2195 ms] 17.45/5.27 (10) BOUNDS(EXP, INF) 17.45/5.27 17.45/5.27 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (0) 17.45/5.27 Obligation: 17.45/5.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.45/5.27 17.45/5.27 17.45/5.27 The TRS R consists of the following rules: 17.45/5.27 17.45/5.27 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 17.45/5.27 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 17.45/5.27 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 17.45/5.27 a__U16(tt) -> tt 17.45/5.27 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 17.45/5.27 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 17.45/5.27 a__U23(tt) -> tt 17.45/5.27 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 17.45/5.27 a__U32(tt) -> tt 17.45/5.27 a__U41(tt) -> tt 17.45/5.27 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 17.45/5.27 a__U52(tt, N) -> mark(N) 17.45/5.27 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 17.45/5.27 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 17.45/5.27 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 17.45/5.27 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 17.45/5.27 a__isNat(0) -> tt 17.45/5.27 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 17.45/5.27 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 17.45/5.27 a__isNatKind(0) -> tt 17.45/5.27 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 17.45/5.27 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 17.45/5.27 a__plus(N, 0) -> a__U51(a__isNat(N), N) 17.45/5.27 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 17.45/5.27 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 17.45/5.27 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 17.45/5.27 mark(isNatKind(X)) -> a__isNatKind(X) 17.45/5.27 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 17.45/5.27 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 17.45/5.27 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 17.45/5.27 mark(isNat(X)) -> a__isNat(X) 17.45/5.27 mark(U16(X)) -> a__U16(mark(X)) 17.45/5.27 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 17.45/5.27 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 17.45/5.27 mark(U23(X)) -> a__U23(mark(X)) 17.45/5.27 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 17.45/5.27 mark(U32(X)) -> a__U32(mark(X)) 17.45/5.27 mark(U41(X)) -> a__U41(mark(X)) 17.45/5.27 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 17.45/5.27 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 17.45/5.27 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 17.45/5.27 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 17.45/5.27 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 17.45/5.27 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 17.45/5.27 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 17.45/5.27 mark(tt) -> tt 17.45/5.27 mark(s(X)) -> s(mark(X)) 17.45/5.27 mark(0) -> 0 17.45/5.27 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 17.45/5.27 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 17.45/5.27 a__isNatKind(X) -> isNatKind(X) 17.45/5.27 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 17.45/5.27 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 17.45/5.27 a__U15(X1, X2) -> U15(X1, X2) 17.45/5.27 a__isNat(X) -> isNat(X) 17.45/5.27 a__U16(X) -> U16(X) 17.45/5.27 a__U21(X1, X2) -> U21(X1, X2) 17.45/5.27 a__U22(X1, X2) -> U22(X1, X2) 17.45/5.27 a__U23(X) -> U23(X) 17.45/5.27 a__U31(X1, X2) -> U31(X1, X2) 17.45/5.27 a__U32(X) -> U32(X) 17.45/5.27 a__U41(X) -> U41(X) 17.45/5.27 a__U51(X1, X2) -> U51(X1, X2) 17.45/5.27 a__U52(X1, X2) -> U52(X1, X2) 17.45/5.27 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 17.45/5.27 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 17.45/5.27 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 17.45/5.27 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 17.45/5.27 a__plus(X1, X2) -> plus(X1, X2) 17.45/5.27 17.45/5.27 S is empty. 17.45/5.27 Rewrite Strategy: FULL 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 17.45/5.27 Transformed a relative TRS into a decreasing-loop problem. 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (2) 17.45/5.27 Obligation: 17.45/5.27 Analyzing the following TRS for decreasing loops: 17.45/5.27 17.45/5.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.45/5.27 17.45/5.27 17.45/5.27 The TRS R consists of the following rules: 17.45/5.27 17.45/5.27 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 17.45/5.27 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 17.45/5.27 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 17.45/5.27 a__U16(tt) -> tt 17.45/5.27 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 17.45/5.27 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 17.45/5.27 a__U23(tt) -> tt 17.45/5.27 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 17.45/5.27 a__U32(tt) -> tt 17.45/5.27 a__U41(tt) -> tt 17.45/5.27 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 17.45/5.27 a__U52(tt, N) -> mark(N) 17.45/5.27 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 17.45/5.27 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 17.45/5.27 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 17.45/5.27 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 17.45/5.27 a__isNat(0) -> tt 17.45/5.27 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 17.45/5.27 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 17.45/5.27 a__isNatKind(0) -> tt 17.45/5.27 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 17.45/5.27 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 17.45/5.27 a__plus(N, 0) -> a__U51(a__isNat(N), N) 17.45/5.27 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 17.45/5.27 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 17.45/5.27 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 17.45/5.27 mark(isNatKind(X)) -> a__isNatKind(X) 17.45/5.27 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 17.45/5.27 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 17.45/5.27 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 17.45/5.27 mark(isNat(X)) -> a__isNat(X) 17.45/5.27 mark(U16(X)) -> a__U16(mark(X)) 17.45/5.27 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 17.45/5.27 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 17.45/5.27 mark(U23(X)) -> a__U23(mark(X)) 17.45/5.27 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 17.45/5.27 mark(U32(X)) -> a__U32(mark(X)) 17.45/5.27 mark(U41(X)) -> a__U41(mark(X)) 17.45/5.27 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 17.45/5.27 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 17.45/5.27 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 17.45/5.27 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 17.45/5.27 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 17.45/5.27 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 17.45/5.27 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 17.45/5.27 mark(tt) -> tt 17.45/5.27 mark(s(X)) -> s(mark(X)) 17.45/5.27 mark(0) -> 0 17.45/5.27 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 17.45/5.27 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 17.45/5.27 a__isNatKind(X) -> isNatKind(X) 17.45/5.27 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 17.45/5.27 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 17.45/5.27 a__U15(X1, X2) -> U15(X1, X2) 17.45/5.27 a__isNat(X) -> isNat(X) 17.45/5.27 a__U16(X) -> U16(X) 17.45/5.27 a__U21(X1, X2) -> U21(X1, X2) 17.45/5.27 a__U22(X1, X2) -> U22(X1, X2) 17.45/5.27 a__U23(X) -> U23(X) 17.45/5.27 a__U31(X1, X2) -> U31(X1, X2) 17.45/5.27 a__U32(X) -> U32(X) 17.45/5.27 a__U41(X) -> U41(X) 17.45/5.27 a__U51(X1, X2) -> U51(X1, X2) 17.45/5.27 a__U52(X1, X2) -> U52(X1, X2) 17.45/5.27 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 17.45/5.27 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 17.45/5.27 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 17.45/5.27 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 17.45/5.27 a__plus(X1, X2) -> plus(X1, X2) 17.45/5.27 17.45/5.27 S is empty. 17.45/5.27 Rewrite Strategy: FULL 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (3) DecreasingLoopProof (LOWER BOUND(ID)) 17.45/5.27 The following loop(s) give(s) rise to the lower bound Omega(n^1): 17.45/5.27 17.45/5.27 The rewrite sequence 17.45/5.27 17.45/5.27 mark(U15(X1, X2)) ->^+ a__U15(mark(X1), X2) 17.45/5.27 17.45/5.27 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 17.45/5.27 17.45/5.27 The pumping substitution is [X1 / U15(X1, X2)]. 17.45/5.27 17.45/5.27 The result substitution is [ ]. 17.45/5.27 17.45/5.27 17.45/5.27 17.45/5.27 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (4) 17.45/5.27 Complex Obligation (BEST) 17.45/5.27 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (5) 17.45/5.27 Obligation: 17.45/5.27 Proved the lower bound n^1 for the following obligation: 17.45/5.27 17.45/5.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.45/5.27 17.45/5.27 17.45/5.27 The TRS R consists of the following rules: 17.45/5.27 17.45/5.27 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 17.45/5.27 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 17.45/5.27 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 17.45/5.27 a__U16(tt) -> tt 17.45/5.27 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 17.45/5.27 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 17.45/5.27 a__U23(tt) -> tt 17.45/5.27 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 17.45/5.27 a__U32(tt) -> tt 17.45/5.27 a__U41(tt) -> tt 17.45/5.27 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 17.45/5.27 a__U52(tt, N) -> mark(N) 17.45/5.27 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 17.45/5.27 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 17.45/5.27 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 17.45/5.27 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 17.45/5.27 a__isNat(0) -> tt 17.45/5.27 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 17.45/5.27 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 17.45/5.27 a__isNatKind(0) -> tt 17.45/5.27 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 17.45/5.27 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 17.45/5.27 a__plus(N, 0) -> a__U51(a__isNat(N), N) 17.45/5.27 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 17.45/5.27 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 17.45/5.27 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 17.45/5.27 mark(isNatKind(X)) -> a__isNatKind(X) 17.45/5.27 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 17.45/5.27 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 17.45/5.27 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 17.45/5.27 mark(isNat(X)) -> a__isNat(X) 17.45/5.27 mark(U16(X)) -> a__U16(mark(X)) 17.45/5.27 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 17.45/5.27 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 17.45/5.27 mark(U23(X)) -> a__U23(mark(X)) 17.45/5.27 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 17.45/5.27 mark(U32(X)) -> a__U32(mark(X)) 17.45/5.27 mark(U41(X)) -> a__U41(mark(X)) 17.45/5.27 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 17.45/5.27 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 17.45/5.27 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 17.45/5.27 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 17.45/5.27 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 17.45/5.27 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 17.45/5.27 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 17.45/5.27 mark(tt) -> tt 17.45/5.27 mark(s(X)) -> s(mark(X)) 17.45/5.27 mark(0) -> 0 17.45/5.27 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 17.45/5.27 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 17.45/5.27 a__isNatKind(X) -> isNatKind(X) 17.45/5.27 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 17.45/5.27 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 17.45/5.27 a__U15(X1, X2) -> U15(X1, X2) 17.45/5.27 a__isNat(X) -> isNat(X) 17.45/5.27 a__U16(X) -> U16(X) 17.45/5.27 a__U21(X1, X2) -> U21(X1, X2) 17.45/5.27 a__U22(X1, X2) -> U22(X1, X2) 17.45/5.27 a__U23(X) -> U23(X) 17.45/5.27 a__U31(X1, X2) -> U31(X1, X2) 17.45/5.27 a__U32(X) -> U32(X) 17.45/5.27 a__U41(X) -> U41(X) 17.45/5.27 a__U51(X1, X2) -> U51(X1, X2) 17.45/5.27 a__U52(X1, X2) -> U52(X1, X2) 17.45/5.27 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 17.45/5.27 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 17.45/5.27 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 17.45/5.27 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 17.45/5.27 a__plus(X1, X2) -> plus(X1, X2) 17.45/5.27 17.45/5.27 S is empty. 17.45/5.27 Rewrite Strategy: FULL 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (6) LowerBoundPropagationProof (FINISHED) 17.45/5.27 Propagated lower bound. 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (7) 17.45/5.27 BOUNDS(n^1, INF) 17.45/5.27 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (8) 17.45/5.27 Obligation: 17.45/5.27 Analyzing the following TRS for decreasing loops: 17.45/5.27 17.45/5.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.45/5.27 17.45/5.27 17.45/5.27 The TRS R consists of the following rules: 17.45/5.27 17.45/5.27 a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) 17.45/5.27 a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) 17.45/5.27 a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) 17.45/5.27 a__U15(tt, V2) -> a__U16(a__isNat(V2)) 17.45/5.27 a__U16(tt) -> tt 17.45/5.27 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 17.45/5.27 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 17.45/5.27 a__U23(tt) -> tt 17.45/5.27 a__U31(tt, V2) -> a__U32(a__isNatKind(V2)) 17.45/5.27 a__U32(tt) -> tt 17.45/5.27 a__U41(tt) -> tt 17.45/5.27 a__U51(tt, N) -> a__U52(a__isNatKind(N), N) 17.45/5.27 a__U52(tt, N) -> mark(N) 17.45/5.27 a__U61(tt, M, N) -> a__U62(a__isNatKind(M), M, N) 17.45/5.27 a__U62(tt, M, N) -> a__U63(a__isNat(N), M, N) 17.45/5.27 a__U63(tt, M, N) -> a__U64(a__isNatKind(N), M, N) 17.45/5.27 a__U64(tt, M, N) -> s(a__plus(mark(N), mark(M))) 17.45/5.27 a__isNat(0) -> tt 17.45/5.27 a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) 17.45/5.27 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 17.45/5.27 a__isNatKind(0) -> tt 17.45/5.27 a__isNatKind(plus(V1, V2)) -> a__U31(a__isNatKind(V1), V2) 17.45/5.27 a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1)) 17.45/5.27 a__plus(N, 0) -> a__U51(a__isNat(N), N) 17.45/5.27 a__plus(N, s(M)) -> a__U61(a__isNat(M), M, N) 17.45/5.27 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 17.45/5.27 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 17.45/5.27 mark(isNatKind(X)) -> a__isNatKind(X) 17.45/5.27 mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) 17.45/5.27 mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) 17.45/5.27 mark(U15(X1, X2)) -> a__U15(mark(X1), X2) 17.45/5.27 mark(isNat(X)) -> a__isNat(X) 17.45/5.27 mark(U16(X)) -> a__U16(mark(X)) 17.45/5.27 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 17.45/5.27 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 17.45/5.27 mark(U23(X)) -> a__U23(mark(X)) 17.45/5.27 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 17.45/5.27 mark(U32(X)) -> a__U32(mark(X)) 17.45/5.27 mark(U41(X)) -> a__U41(mark(X)) 17.45/5.27 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 17.45/5.27 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 17.45/5.27 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 17.45/5.27 mark(U62(X1, X2, X3)) -> a__U62(mark(X1), X2, X3) 17.45/5.27 mark(U63(X1, X2, X3)) -> a__U63(mark(X1), X2, X3) 17.45/5.27 mark(U64(X1, X2, X3)) -> a__U64(mark(X1), X2, X3) 17.45/5.27 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 17.45/5.27 mark(tt) -> tt 17.45/5.27 mark(s(X)) -> s(mark(X)) 17.45/5.27 mark(0) -> 0 17.45/5.27 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 17.45/5.27 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 17.45/5.27 a__isNatKind(X) -> isNatKind(X) 17.45/5.27 a__U13(X1, X2, X3) -> U13(X1, X2, X3) 17.45/5.27 a__U14(X1, X2, X3) -> U14(X1, X2, X3) 17.45/5.27 a__U15(X1, X2) -> U15(X1, X2) 17.45/5.27 a__isNat(X) -> isNat(X) 17.45/5.27 a__U16(X) -> U16(X) 17.45/5.27 a__U21(X1, X2) -> U21(X1, X2) 17.45/5.27 a__U22(X1, X2) -> U22(X1, X2) 17.45/5.27 a__U23(X) -> U23(X) 17.45/5.27 a__U31(X1, X2) -> U31(X1, X2) 17.45/5.27 a__U32(X) -> U32(X) 17.45/5.27 a__U41(X) -> U41(X) 17.45/5.27 a__U51(X1, X2) -> U51(X1, X2) 17.45/5.27 a__U52(X1, X2) -> U52(X1, X2) 17.45/5.27 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 17.45/5.27 a__U62(X1, X2, X3) -> U62(X1, X2, X3) 17.45/5.27 a__U63(X1, X2, X3) -> U63(X1, X2, X3) 17.45/5.27 a__U64(X1, X2, X3) -> U64(X1, X2, X3) 17.45/5.27 a__plus(X1, X2) -> plus(X1, X2) 17.45/5.27 17.45/5.27 S is empty. 17.45/5.27 Rewrite Strategy: FULL 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (9) DecreasingLoopProof (FINISHED) 17.45/5.27 The following loop(s) give(s) rise to the lower bound EXP: 17.45/5.27 17.45/5.27 The rewrite sequence 17.45/5.27 17.45/5.27 mark(plus(X1, s(X1_0))) ->^+ a__U61(a__isNat(mark(X1_0)), mark(X1_0), mark(X1)) 17.45/5.27 17.45/5.27 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 17.45/5.27 17.45/5.27 The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. 17.45/5.27 17.45/5.27 The result substitution is [ ]. 17.45/5.27 17.45/5.27 17.45/5.27 17.45/5.27 The rewrite sequence 17.45/5.27 17.45/5.27 mark(plus(X1, s(X1_0))) ->^+ a__U61(a__isNat(mark(X1_0)), mark(X1_0), mark(X1)) 17.45/5.27 17.45/5.27 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 17.45/5.27 17.45/5.27 The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. 17.45/5.27 17.45/5.27 The result substitution is [ ]. 17.45/5.27 17.45/5.27 17.45/5.27 17.45/5.27 17.45/5.27 ---------------------------------------- 17.45/5.27 17.45/5.27 (10) 17.45/5.27 BOUNDS(EXP, INF) 17.69/5.34 EOF