63.12/19.51 WORST_CASE(NON_POLY, ?) 63.39/19.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 63.39/19.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 63.39/19.52 63.39/19.52 63.39/19.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 63.39/19.52 63.39/19.52 (0) CpxTRS 63.39/19.52 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 63.39/19.52 (2) TRS for Loop Detection 63.39/19.52 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 63.39/19.52 (4) BEST 63.39/19.52 (5) proven lower bound 63.39/19.52 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 63.39/19.52 (7) BOUNDS(n^1, INF) 63.39/19.52 (8) TRS for Loop Detection 63.39/19.52 (9) DecreasingLoopProof [FINISHED, 15.9 s] 63.39/19.52 (10) BOUNDS(EXP, INF) 63.39/19.52 63.39/19.52 63.39/19.52 ---------------------------------------- 63.39/19.52 63.39/19.52 (0) 63.39/19.52 Obligation: 63.39/19.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 63.39/19.52 63.39/19.52 63.39/19.52 The TRS R consists of the following rules: 63.39/19.52 63.39/19.52 a__zeros -> cons(0, zeros) 63.39/19.52 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 63.39/19.52 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 63.39/19.52 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 63.39/19.52 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 63.39/19.52 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 63.39/19.52 a__U106(tt) -> tt 63.39/19.52 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 63.39/19.52 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 63.39/19.52 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 63.39/19.52 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 63.39/19.52 a__U114(tt, L) -> s(a__length(mark(L))) 63.39/19.52 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 63.39/19.52 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 63.39/19.52 a__U122(tt) -> nil 63.39/19.52 a__U13(tt) -> tt 63.39/19.52 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 63.39/19.52 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 63.39/19.52 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 63.39/19.52 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 63.39/19.52 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 63.39/19.52 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 63.39/19.52 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 63.39/19.52 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 63.39/19.52 a__U23(tt) -> tt 63.39/19.52 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 63.39/19.52 a__U32(tt, V) -> a__U33(a__isNatList(V)) 63.39/19.52 a__U33(tt) -> tt 63.39/19.52 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 63.39/19.52 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 63.39/19.52 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 63.39/19.52 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 63.39/19.52 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 63.39/19.52 a__U46(tt) -> tt 63.39/19.52 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 63.39/19.52 a__U52(tt) -> tt 63.39/19.52 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 63.39/19.52 a__U62(tt) -> tt 63.39/19.52 a__U71(tt) -> tt 63.39/19.52 a__U81(tt) -> tt 63.39/19.52 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 63.39/19.52 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 63.39/19.52 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 63.39/19.52 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 63.39/19.52 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 63.39/19.52 a__U96(tt) -> tt 63.39/19.52 a__isNat(0) -> tt 63.39/19.52 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 63.39/19.52 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 63.39/19.52 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 63.39/19.52 a__isNatIList(zeros) -> tt 63.39/19.52 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 63.39/19.52 a__isNatIListKind(nil) -> tt 63.39/19.52 a__isNatIListKind(zeros) -> tt 63.39/19.52 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 63.39/19.52 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 63.39/19.52 a__isNatKind(0) -> tt 63.39/19.52 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 63.39/19.52 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 63.39/19.52 a__isNatList(nil) -> tt 63.39/19.52 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 63.39/19.52 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 63.39/19.52 a__length(nil) -> 0 63.39/19.52 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 63.39/19.52 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 63.39/19.52 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 63.39/19.52 mark(zeros) -> a__zeros 63.39/19.52 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 63.39/19.52 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 63.39/19.52 mark(isNatKind(X)) -> a__isNatKind(X) 63.39/19.52 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 63.39/19.52 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 63.39/19.52 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 63.39/19.52 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 63.39/19.52 mark(isNat(X)) -> a__isNat(X) 63.39/19.52 mark(U106(X)) -> a__U106(mark(X)) 63.39/19.52 mark(isNatIList(X)) -> a__isNatIList(X) 63.39/19.52 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 63.39/19.52 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 63.39/19.52 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 63.39/19.52 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 63.39/19.52 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 63.39/19.52 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 63.39/19.52 mark(length(X)) -> a__length(mark(X)) 63.39/19.52 mark(U13(X)) -> a__U13(mark(X)) 63.39/19.52 mark(isNatList(X)) -> a__isNatList(X) 63.39/19.52 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 63.39/19.52 mark(U122(X)) -> a__U122(mark(X)) 63.39/19.52 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 63.39/19.52 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 63.39/19.52 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 63.39/19.52 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 63.39/19.52 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 63.39/19.52 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 63.39/19.52 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 63.39/19.52 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 63.39/19.52 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 63.39/19.52 mark(U23(X)) -> a__U23(mark(X)) 63.39/19.52 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 63.39/19.52 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 63.39/19.52 mark(U33(X)) -> a__U33(mark(X)) 63.39/19.52 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 63.39/19.52 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 63.39/19.52 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 63.39/19.52 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 63.39/19.52 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 63.39/19.52 mark(U46(X)) -> a__U46(mark(X)) 63.39/19.52 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 63.39/19.52 mark(U52(X)) -> a__U52(mark(X)) 63.39/19.52 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 63.39/19.52 mark(U62(X)) -> a__U62(mark(X)) 63.39/19.52 mark(U71(X)) -> a__U71(mark(X)) 63.39/19.52 mark(U81(X)) -> a__U81(mark(X)) 63.39/19.52 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 63.39/19.52 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 63.39/19.52 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 63.39/19.52 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 63.39/19.52 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 63.39/19.52 mark(U96(X)) -> a__U96(mark(X)) 63.39/19.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 63.39/19.52 mark(0) -> 0 63.39/19.52 mark(tt) -> tt 63.39/19.52 mark(s(X)) -> s(mark(X)) 63.39/19.52 mark(nil) -> nil 63.39/19.52 a__zeros -> zeros 63.39/19.52 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 63.39/19.52 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 63.39/19.52 a__isNatKind(X) -> isNatKind(X) 63.39/19.52 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 63.39/19.52 a__isNatIListKind(X) -> isNatIListKind(X) 63.39/19.52 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 63.39/19.52 a__U105(X1, X2) -> U105(X1, X2) 63.39/19.52 a__isNat(X) -> isNat(X) 63.39/19.52 a__U106(X) -> U106(X) 63.39/19.52 a__isNatIList(X) -> isNatIList(X) 63.39/19.52 a__U11(X1, X2) -> U11(X1, X2) 63.39/19.52 a__U12(X1, X2) -> U12(X1, X2) 63.39/19.52 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 63.39/19.52 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 63.39/19.52 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 63.39/19.52 a__U114(X1, X2) -> U114(X1, X2) 63.39/19.53 a__length(X) -> length(X) 63.39/19.53 a__U13(X) -> U13(X) 63.39/19.53 a__isNatList(X) -> isNatList(X) 63.39/19.53 a__U121(X1, X2) -> U121(X1, X2) 63.39/19.53 a__U122(X) -> U122(X) 63.39/19.53 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 63.39/19.53 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 63.39/19.53 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 63.39/19.53 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 63.39/19.53 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 63.39/19.53 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 63.39/19.53 a__take(X1, X2) -> take(X1, X2) 63.39/19.53 a__U21(X1, X2) -> U21(X1, X2) 63.39/19.53 a__U22(X1, X2) -> U22(X1, X2) 63.39/19.53 a__U23(X) -> U23(X) 63.39/19.53 a__U31(X1, X2) -> U31(X1, X2) 63.39/19.53 a__U32(X1, X2) -> U32(X1, X2) 63.39/19.53 a__U33(X) -> U33(X) 63.39/19.53 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 63.39/19.53 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 63.39/19.53 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 63.39/19.53 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 63.39/19.53 a__U45(X1, X2) -> U45(X1, X2) 63.39/19.53 a__U46(X) -> U46(X) 63.39/19.53 a__U51(X1, X2) -> U51(X1, X2) 63.39/19.53 a__U52(X) -> U52(X) 63.39/19.53 a__U61(X1, X2) -> U61(X1, X2) 63.39/19.53 a__U62(X) -> U62(X) 63.39/19.53 a__U71(X) -> U71(X) 63.39/19.53 a__U81(X) -> U81(X) 63.39/19.53 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 63.39/19.53 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 63.39/19.53 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 63.39/19.53 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 63.39/19.53 a__U95(X1, X2) -> U95(X1, X2) 63.39/19.53 a__U96(X) -> U96(X) 63.39/19.53 63.39/19.53 S is empty. 63.39/19.53 Rewrite Strategy: FULL 63.39/19.53 ---------------------------------------- 63.39/19.53 63.39/19.53 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 63.39/19.53 Transformed a relative TRS into a decreasing-loop problem. 63.39/19.53 ---------------------------------------- 63.39/19.53 63.39/19.53 (2) 63.39/19.53 Obligation: 63.39/19.53 Analyzing the following TRS for decreasing loops: 63.39/19.53 63.39/19.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 63.39/19.53 63.39/19.53 63.39/19.53 The TRS R consists of the following rules: 63.39/19.53 63.39/19.53 a__zeros -> cons(0, zeros) 63.39/19.53 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 63.39/19.53 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 63.39/19.53 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 63.39/19.53 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 63.39/19.53 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 63.39/19.53 a__U106(tt) -> tt 63.39/19.53 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 63.39/19.53 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 63.39/19.53 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 63.39/19.53 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 63.39/19.53 a__U114(tt, L) -> s(a__length(mark(L))) 63.39/19.53 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 63.39/19.53 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 63.39/19.53 a__U122(tt) -> nil 63.39/19.53 a__U13(tt) -> tt 63.39/19.53 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 63.39/19.53 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 63.39/19.53 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 63.39/19.53 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 63.39/19.53 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 63.39/19.53 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 63.39/19.53 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 63.39/19.53 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 63.39/19.53 a__U23(tt) -> tt 63.39/19.53 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 63.39/19.53 a__U32(tt, V) -> a__U33(a__isNatList(V)) 63.39/19.53 a__U33(tt) -> tt 63.39/19.53 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 63.39/19.53 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 63.39/19.53 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 63.39/19.53 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 63.39/19.53 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 63.39/19.53 a__U46(tt) -> tt 63.39/19.53 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 63.39/19.53 a__U52(tt) -> tt 63.39/19.53 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 63.39/19.53 a__U62(tt) -> tt 63.39/19.53 a__U71(tt) -> tt 63.39/19.53 a__U81(tt) -> tt 63.39/19.53 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 63.39/19.53 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 63.39/19.53 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 63.39/19.53 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 63.39/19.53 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 63.39/19.53 a__U96(tt) -> tt 63.39/19.53 a__isNat(0) -> tt 63.39/19.53 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 63.39/19.53 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 63.39/19.53 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 63.39/19.53 a__isNatIList(zeros) -> tt 63.39/19.53 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 63.39/19.53 a__isNatIListKind(nil) -> tt 63.39/19.53 a__isNatIListKind(zeros) -> tt 63.39/19.53 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 63.39/19.53 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 63.39/19.53 a__isNatKind(0) -> tt 63.39/19.53 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 63.39/19.53 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 63.39/19.53 a__isNatList(nil) -> tt 63.39/19.53 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 63.39/19.53 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 63.39/19.53 a__length(nil) -> 0 63.39/19.53 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 63.39/19.53 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 63.39/19.53 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 63.39/19.53 mark(zeros) -> a__zeros 63.39/19.53 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 63.39/19.53 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 63.39/19.53 mark(isNatKind(X)) -> a__isNatKind(X) 63.39/19.53 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 63.39/19.53 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 63.39/19.53 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 63.39/19.53 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 63.39/19.53 mark(isNat(X)) -> a__isNat(X) 63.39/19.53 mark(U106(X)) -> a__U106(mark(X)) 63.39/19.53 mark(isNatIList(X)) -> a__isNatIList(X) 63.39/19.53 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 63.39/19.53 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 63.39/19.53 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 63.39/19.53 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 63.39/19.53 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 63.39/19.53 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 63.39/19.53 mark(length(X)) -> a__length(mark(X)) 63.39/19.53 mark(U13(X)) -> a__U13(mark(X)) 63.39/19.53 mark(isNatList(X)) -> a__isNatList(X) 63.39/19.53 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 63.39/19.53 mark(U122(X)) -> a__U122(mark(X)) 63.39/19.53 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 63.39/19.53 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 63.39/19.53 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 63.39/19.53 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 63.39/19.53 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 63.39/19.53 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 63.39/19.53 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 63.39/19.53 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 63.39/19.53 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 63.39/19.53 mark(U23(X)) -> a__U23(mark(X)) 63.39/19.53 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 63.39/19.53 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 63.39/19.53 mark(U33(X)) -> a__U33(mark(X)) 63.39/19.53 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 63.39/19.53 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 63.39/19.53 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 63.39/19.53 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 63.39/19.53 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 63.39/19.53 mark(U46(X)) -> a__U46(mark(X)) 63.39/19.53 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 63.39/19.53 mark(U52(X)) -> a__U52(mark(X)) 63.39/19.53 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 63.39/19.53 mark(U62(X)) -> a__U62(mark(X)) 63.39/19.53 mark(U71(X)) -> a__U71(mark(X)) 63.39/19.53 mark(U81(X)) -> a__U81(mark(X)) 63.39/19.53 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 63.39/19.53 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 63.39/19.53 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 63.39/19.53 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 63.39/19.53 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 63.39/19.53 mark(U96(X)) -> a__U96(mark(X)) 63.39/19.53 mark(cons(X1, X2)) -> cons(mark(X1), X2) 63.39/19.53 mark(0) -> 0 63.39/19.53 mark(tt) -> tt 63.39/19.53 mark(s(X)) -> s(mark(X)) 63.39/19.53 mark(nil) -> nil 63.39/19.53 a__zeros -> zeros 63.39/19.53 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 63.39/19.53 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 63.39/19.53 a__isNatKind(X) -> isNatKind(X) 63.39/19.53 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 63.39/19.53 a__isNatIListKind(X) -> isNatIListKind(X) 63.39/19.53 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 63.39/19.53 a__U105(X1, X2) -> U105(X1, X2) 63.39/19.53 a__isNat(X) -> isNat(X) 63.39/19.53 a__U106(X) -> U106(X) 63.39/19.53 a__isNatIList(X) -> isNatIList(X) 63.39/19.53 a__U11(X1, X2) -> U11(X1, X2) 63.39/19.53 a__U12(X1, X2) -> U12(X1, X2) 63.39/19.53 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 63.39/19.53 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 63.39/19.53 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 63.39/19.53 a__U114(X1, X2) -> U114(X1, X2) 63.39/19.53 a__length(X) -> length(X) 63.39/19.53 a__U13(X) -> U13(X) 63.39/19.53 a__isNatList(X) -> isNatList(X) 63.39/19.53 a__U121(X1, X2) -> U121(X1, X2) 63.39/19.53 a__U122(X) -> U122(X) 63.39/19.53 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 63.39/19.53 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 63.39/19.53 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 63.39/19.53 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 63.39/19.53 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 63.39/19.53 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 63.39/19.53 a__take(X1, X2) -> take(X1, X2) 63.39/19.53 a__U21(X1, X2) -> U21(X1, X2) 63.39/19.53 a__U22(X1, X2) -> U22(X1, X2) 63.39/19.53 a__U23(X) -> U23(X) 63.39/19.53 a__U31(X1, X2) -> U31(X1, X2) 63.39/19.53 a__U32(X1, X2) -> U32(X1, X2) 63.39/19.53 a__U33(X) -> U33(X) 63.39/19.53 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 63.39/19.53 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 63.39/19.53 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 63.39/19.53 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 63.39/19.53 a__U45(X1, X2) -> U45(X1, X2) 63.39/19.53 a__U46(X) -> U46(X) 63.39/19.53 a__U51(X1, X2) -> U51(X1, X2) 63.39/19.53 a__U52(X) -> U52(X) 63.39/19.53 a__U61(X1, X2) -> U61(X1, X2) 63.39/19.53 a__U62(X) -> U62(X) 63.39/19.53 a__U71(X) -> U71(X) 63.39/19.53 a__U81(X) -> U81(X) 63.39/19.53 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 63.39/19.53 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 63.39/19.53 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 63.39/19.54 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 63.39/19.54 a__U95(X1, X2) -> U95(X1, X2) 63.39/19.54 a__U96(X) -> U96(X) 63.39/19.54 63.39/19.54 S is empty. 63.39/19.54 Rewrite Strategy: FULL 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (3) DecreasingLoopProof (LOWER BOUND(ID)) 63.39/19.54 The following loop(s) give(s) rise to the lower bound Omega(n^1): 63.39/19.54 63.39/19.54 The rewrite sequence 63.39/19.54 63.39/19.54 mark(U71(X)) ->^+ a__U71(mark(X)) 63.39/19.54 63.39/19.54 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 63.39/19.54 63.39/19.54 The pumping substitution is [X / U71(X)]. 63.39/19.54 63.39/19.54 The result substitution is [ ]. 63.39/19.54 63.39/19.54 63.39/19.54 63.39/19.54 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (4) 63.39/19.54 Complex Obligation (BEST) 63.39/19.54 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (5) 63.39/19.54 Obligation: 63.39/19.54 Proved the lower bound n^1 for the following obligation: 63.39/19.54 63.39/19.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 63.39/19.54 63.39/19.54 63.39/19.54 The TRS R consists of the following rules: 63.39/19.54 63.39/19.54 a__zeros -> cons(0, zeros) 63.39/19.54 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 63.39/19.54 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 63.39/19.54 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 63.39/19.54 a__U106(tt) -> tt 63.39/19.54 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 63.39/19.54 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 63.39/19.54 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 63.39/19.54 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 63.39/19.54 a__U114(tt, L) -> s(a__length(mark(L))) 63.39/19.54 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 63.39/19.54 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 63.39/19.54 a__U122(tt) -> nil 63.39/19.54 a__U13(tt) -> tt 63.39/19.54 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 63.39/19.54 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 63.39/19.54 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 63.39/19.54 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 63.39/19.54 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 63.39/19.54 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 63.39/19.54 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 63.39/19.54 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 63.39/19.54 a__U23(tt) -> tt 63.39/19.54 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 63.39/19.54 a__U32(tt, V) -> a__U33(a__isNatList(V)) 63.39/19.54 a__U33(tt) -> tt 63.39/19.54 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 63.39/19.54 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 63.39/19.54 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 63.39/19.54 a__U46(tt) -> tt 63.39/19.54 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 63.39/19.54 a__U52(tt) -> tt 63.39/19.54 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 63.39/19.54 a__U62(tt) -> tt 63.39/19.54 a__U71(tt) -> tt 63.39/19.54 a__U81(tt) -> tt 63.39/19.54 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 63.39/19.54 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 63.39/19.54 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 63.39/19.54 a__U96(tt) -> tt 63.39/19.54 a__isNat(0) -> tt 63.39/19.54 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 63.39/19.54 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 63.39/19.54 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 63.39/19.54 a__isNatIList(zeros) -> tt 63.39/19.54 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 63.39/19.54 a__isNatIListKind(nil) -> tt 63.39/19.54 a__isNatIListKind(zeros) -> tt 63.39/19.54 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 63.39/19.54 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 63.39/19.54 a__isNatKind(0) -> tt 63.39/19.54 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 63.39/19.54 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 63.39/19.54 a__isNatList(nil) -> tt 63.39/19.54 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 63.39/19.54 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 63.39/19.54 a__length(nil) -> 0 63.39/19.54 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 63.39/19.54 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 63.39/19.54 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 63.39/19.54 mark(zeros) -> a__zeros 63.39/19.54 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 63.39/19.54 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 63.39/19.54 mark(isNatKind(X)) -> a__isNatKind(X) 63.39/19.54 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 63.39/19.54 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 63.39/19.54 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 63.39/19.54 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 63.39/19.54 mark(isNat(X)) -> a__isNat(X) 63.39/19.54 mark(U106(X)) -> a__U106(mark(X)) 63.39/19.54 mark(isNatIList(X)) -> a__isNatIList(X) 63.39/19.54 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 63.39/19.54 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 63.39/19.54 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 63.39/19.54 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 63.39/19.54 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 63.39/19.54 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 63.39/19.54 mark(length(X)) -> a__length(mark(X)) 63.39/19.54 mark(U13(X)) -> a__U13(mark(X)) 63.39/19.54 mark(isNatList(X)) -> a__isNatList(X) 63.39/19.54 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 63.39/19.54 mark(U122(X)) -> a__U122(mark(X)) 63.39/19.54 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 63.39/19.54 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 63.39/19.54 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 63.39/19.54 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 63.39/19.54 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 63.39/19.54 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 63.39/19.54 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 63.39/19.54 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 63.39/19.54 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 63.39/19.54 mark(U23(X)) -> a__U23(mark(X)) 63.39/19.54 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 63.39/19.54 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 63.39/19.54 mark(U33(X)) -> a__U33(mark(X)) 63.39/19.54 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 63.39/19.54 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 63.39/19.54 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 63.39/19.54 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 63.39/19.54 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 63.39/19.54 mark(U46(X)) -> a__U46(mark(X)) 63.39/19.54 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 63.39/19.54 mark(U52(X)) -> a__U52(mark(X)) 63.39/19.54 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 63.39/19.54 mark(U62(X)) -> a__U62(mark(X)) 63.39/19.54 mark(U71(X)) -> a__U71(mark(X)) 63.39/19.54 mark(U81(X)) -> a__U81(mark(X)) 63.39/19.54 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 63.39/19.54 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 63.39/19.54 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 63.39/19.54 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 63.39/19.54 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 63.39/19.54 mark(U96(X)) -> a__U96(mark(X)) 63.39/19.54 mark(cons(X1, X2)) -> cons(mark(X1), X2) 63.39/19.54 mark(0) -> 0 63.39/19.54 mark(tt) -> tt 63.39/19.54 mark(s(X)) -> s(mark(X)) 63.39/19.54 mark(nil) -> nil 63.39/19.54 a__zeros -> zeros 63.39/19.54 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 63.39/19.54 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 63.39/19.54 a__isNatKind(X) -> isNatKind(X) 63.39/19.54 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 63.39/19.54 a__isNatIListKind(X) -> isNatIListKind(X) 63.39/19.54 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 63.39/19.54 a__U105(X1, X2) -> U105(X1, X2) 63.39/19.54 a__isNat(X) -> isNat(X) 63.39/19.54 a__U106(X) -> U106(X) 63.39/19.54 a__isNatIList(X) -> isNatIList(X) 63.39/19.54 a__U11(X1, X2) -> U11(X1, X2) 63.39/19.54 a__U12(X1, X2) -> U12(X1, X2) 63.39/19.54 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 63.39/19.54 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 63.39/19.54 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 63.39/19.54 a__U114(X1, X2) -> U114(X1, X2) 63.39/19.54 a__length(X) -> length(X) 63.39/19.54 a__U13(X) -> U13(X) 63.39/19.54 a__isNatList(X) -> isNatList(X) 63.39/19.54 a__U121(X1, X2) -> U121(X1, X2) 63.39/19.54 a__U122(X) -> U122(X) 63.39/19.54 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 63.39/19.54 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 63.39/19.54 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 63.39/19.54 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 63.39/19.54 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 63.39/19.54 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 63.39/19.54 a__take(X1, X2) -> take(X1, X2) 63.39/19.54 a__U21(X1, X2) -> U21(X1, X2) 63.39/19.54 a__U22(X1, X2) -> U22(X1, X2) 63.39/19.54 a__U23(X) -> U23(X) 63.39/19.54 a__U31(X1, X2) -> U31(X1, X2) 63.39/19.54 a__U32(X1, X2) -> U32(X1, X2) 63.39/19.54 a__U33(X) -> U33(X) 63.39/19.54 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 63.39/19.54 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 63.39/19.54 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 63.39/19.54 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 63.39/19.54 a__U45(X1, X2) -> U45(X1, X2) 63.39/19.54 a__U46(X) -> U46(X) 63.39/19.54 a__U51(X1, X2) -> U51(X1, X2) 63.39/19.54 a__U52(X) -> U52(X) 63.39/19.54 a__U61(X1, X2) -> U61(X1, X2) 63.39/19.54 a__U62(X) -> U62(X) 63.39/19.54 a__U71(X) -> U71(X) 63.39/19.54 a__U81(X) -> U81(X) 63.39/19.54 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 63.39/19.54 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 63.39/19.54 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 63.39/19.54 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 63.39/19.54 a__U95(X1, X2) -> U95(X1, X2) 63.39/19.54 a__U96(X) -> U96(X) 63.39/19.54 63.39/19.54 S is empty. 63.39/19.54 Rewrite Strategy: FULL 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (6) LowerBoundPropagationProof (FINISHED) 63.39/19.54 Propagated lower bound. 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (7) 63.39/19.54 BOUNDS(n^1, INF) 63.39/19.54 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (8) 63.39/19.54 Obligation: 63.39/19.54 Analyzing the following TRS for decreasing loops: 63.39/19.54 63.39/19.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 63.39/19.54 63.39/19.54 63.39/19.54 The TRS R consists of the following rules: 63.39/19.54 63.39/19.54 a__zeros -> cons(0, zeros) 63.39/19.54 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 63.39/19.54 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 63.39/19.54 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 63.39/19.54 a__U106(tt) -> tt 63.39/19.54 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 63.39/19.54 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 63.39/19.54 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 63.39/19.54 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 63.39/19.54 a__U114(tt, L) -> s(a__length(mark(L))) 63.39/19.54 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 63.39/19.54 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 63.39/19.54 a__U122(tt) -> nil 63.39/19.54 a__U13(tt) -> tt 63.39/19.54 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 63.39/19.54 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 63.39/19.54 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 63.39/19.54 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 63.39/19.54 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 63.39/19.54 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 63.39/19.54 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 63.39/19.54 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 63.39/19.54 a__U23(tt) -> tt 63.39/19.54 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 63.39/19.54 a__U32(tt, V) -> a__U33(a__isNatList(V)) 63.39/19.54 a__U33(tt) -> tt 63.39/19.54 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 63.39/19.54 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 63.39/19.54 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 63.39/19.54 a__U46(tt) -> tt 63.39/19.54 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 63.39/19.54 a__U52(tt) -> tt 63.39/19.54 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 63.39/19.54 a__U62(tt) -> tt 63.39/19.54 a__U71(tt) -> tt 63.39/19.54 a__U81(tt) -> tt 63.39/19.54 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 63.39/19.54 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 63.39/19.54 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 63.39/19.54 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 63.39/19.54 a__U96(tt) -> tt 63.39/19.54 a__isNat(0) -> tt 63.39/19.54 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 63.39/19.54 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 63.39/19.54 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 63.39/19.54 a__isNatIList(zeros) -> tt 63.39/19.54 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 63.39/19.54 a__isNatIListKind(nil) -> tt 63.39/19.54 a__isNatIListKind(zeros) -> tt 63.39/19.54 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 63.39/19.54 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 63.39/19.54 a__isNatKind(0) -> tt 63.39/19.54 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 63.39/19.54 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 63.39/19.54 a__isNatList(nil) -> tt 63.39/19.54 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 63.39/19.54 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 63.39/19.54 a__length(nil) -> 0 63.39/19.54 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 63.39/19.54 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 63.39/19.54 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 63.39/19.54 mark(zeros) -> a__zeros 63.39/19.54 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 63.39/19.54 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 63.39/19.54 mark(isNatKind(X)) -> a__isNatKind(X) 63.39/19.54 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 63.39/19.54 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 63.39/19.54 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 63.39/19.54 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 63.39/19.54 mark(isNat(X)) -> a__isNat(X) 63.39/19.54 mark(U106(X)) -> a__U106(mark(X)) 63.39/19.54 mark(isNatIList(X)) -> a__isNatIList(X) 63.39/19.54 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 63.39/19.54 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 63.39/19.54 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 63.39/19.54 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 63.39/19.54 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 63.39/19.54 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 63.39/19.54 mark(length(X)) -> a__length(mark(X)) 63.39/19.54 mark(U13(X)) -> a__U13(mark(X)) 63.39/19.54 mark(isNatList(X)) -> a__isNatList(X) 63.39/19.54 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 63.39/19.54 mark(U122(X)) -> a__U122(mark(X)) 63.39/19.54 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 63.39/19.54 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 63.39/19.54 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 63.39/19.54 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 63.39/19.54 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 63.39/19.54 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 63.39/19.54 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 63.39/19.54 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 63.39/19.54 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 63.39/19.54 mark(U23(X)) -> a__U23(mark(X)) 63.39/19.54 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 63.39/19.54 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 63.39/19.54 mark(U33(X)) -> a__U33(mark(X)) 63.39/19.54 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 63.39/19.54 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 63.39/19.54 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 63.39/19.54 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 63.39/19.54 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 63.39/19.54 mark(U46(X)) -> a__U46(mark(X)) 63.39/19.54 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 63.39/19.54 mark(U52(X)) -> a__U52(mark(X)) 63.39/19.54 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 63.39/19.54 mark(U62(X)) -> a__U62(mark(X)) 63.39/19.54 mark(U71(X)) -> a__U71(mark(X)) 63.39/19.54 mark(U81(X)) -> a__U81(mark(X)) 63.39/19.54 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 63.39/19.54 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 63.39/19.54 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 63.39/19.54 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 63.39/19.54 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 63.39/19.54 mark(U96(X)) -> a__U96(mark(X)) 63.39/19.54 mark(cons(X1, X2)) -> cons(mark(X1), X2) 63.39/19.54 mark(0) -> 0 63.39/19.54 mark(tt) -> tt 63.39/19.54 mark(s(X)) -> s(mark(X)) 63.39/19.54 mark(nil) -> nil 63.39/19.54 a__zeros -> zeros 63.39/19.54 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 63.39/19.54 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 63.39/19.54 a__isNatKind(X) -> isNatKind(X) 63.39/19.54 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 63.39/19.54 a__isNatIListKind(X) -> isNatIListKind(X) 63.39/19.54 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 63.39/19.54 a__U105(X1, X2) -> U105(X1, X2) 63.39/19.54 a__isNat(X) -> isNat(X) 63.39/19.54 a__U106(X) -> U106(X) 63.39/19.54 a__isNatIList(X) -> isNatIList(X) 63.39/19.54 a__U11(X1, X2) -> U11(X1, X2) 63.39/19.54 a__U12(X1, X2) -> U12(X1, X2) 63.39/19.54 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 63.39/19.54 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 63.39/19.54 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 63.39/19.54 a__U114(X1, X2) -> U114(X1, X2) 63.39/19.54 a__length(X) -> length(X) 63.39/19.54 a__U13(X) -> U13(X) 63.39/19.54 a__isNatList(X) -> isNatList(X) 63.39/19.54 a__U121(X1, X2) -> U121(X1, X2) 63.39/19.54 a__U122(X) -> U122(X) 63.39/19.54 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 63.39/19.54 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 63.39/19.54 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 63.39/19.54 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 63.39/19.54 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 63.39/19.54 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 63.39/19.54 a__take(X1, X2) -> take(X1, X2) 63.39/19.54 a__U21(X1, X2) -> U21(X1, X2) 63.39/19.54 a__U22(X1, X2) -> U22(X1, X2) 63.39/19.54 a__U23(X) -> U23(X) 63.39/19.54 a__U31(X1, X2) -> U31(X1, X2) 63.39/19.54 a__U32(X1, X2) -> U32(X1, X2) 63.39/19.54 a__U33(X) -> U33(X) 63.39/19.54 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 63.39/19.54 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 63.39/19.54 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 63.39/19.54 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 63.39/19.54 a__U45(X1, X2) -> U45(X1, X2) 63.39/19.54 a__U46(X) -> U46(X) 63.39/19.54 a__U51(X1, X2) -> U51(X1, X2) 63.39/19.54 a__U52(X) -> U52(X) 63.39/19.54 a__U61(X1, X2) -> U61(X1, X2) 63.39/19.54 a__U62(X) -> U62(X) 63.39/19.54 a__U71(X) -> U71(X) 63.39/19.54 a__U81(X) -> U81(X) 63.39/19.54 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 63.39/19.54 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 63.39/19.54 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 63.39/19.54 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 63.39/19.54 a__U95(X1, X2) -> U95(X1, X2) 63.39/19.54 a__U96(X) -> U96(X) 63.39/19.54 63.39/19.54 S is empty. 63.39/19.54 Rewrite Strategy: FULL 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (9) DecreasingLoopProof (FINISHED) 63.39/19.54 The following loop(s) give(s) rise to the lower bound EXP: 63.39/19.54 63.39/19.54 The rewrite sequence 63.39/19.54 63.39/19.54 mark(take(0, X2)) ->^+ a__U121(a__isNatIList(mark(X2)), mark(X2)) 63.39/19.54 63.39/19.54 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 63.39/19.54 63.39/19.54 The pumping substitution is [X2 / take(0, X2)]. 63.39/19.54 63.39/19.54 The result substitution is [ ]. 63.39/19.54 63.39/19.54 63.39/19.54 63.39/19.54 The rewrite sequence 63.39/19.54 63.39/19.54 mark(take(0, X2)) ->^+ a__U121(a__isNatIList(mark(X2)), mark(X2)) 63.39/19.54 63.39/19.54 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 63.39/19.54 63.39/19.54 The pumping substitution is [X2 / take(0, X2)]. 63.39/19.54 63.39/19.54 The result substitution is [ ]. 63.39/19.54 63.39/19.54 63.39/19.54 63.39/19.54 63.39/19.54 ---------------------------------------- 63.39/19.54 63.39/19.54 (10) 63.39/19.54 BOUNDS(EXP, INF) 63.51/19.61 EOF