1146.86/291.59 WORST_CASE(Omega(n^1), ?) 1160.44/294.97 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1160.44/294.97 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1160.44/294.97 1160.44/294.97 1160.44/294.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1160.44/294.97 1160.44/294.97 (0) CpxTRS 1160.44/294.97 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1160.44/294.97 (2) TRS for Loop Detection 1160.44/294.97 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1160.44/294.97 (4) BEST 1160.44/294.97 (5) proven lower bound 1160.44/294.97 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1160.44/294.97 (7) BOUNDS(n^1, INF) 1160.44/294.97 (8) TRS for Loop Detection 1160.44/294.97 1160.44/294.97 1160.44/294.97 ---------------------------------------- 1160.44/294.97 1160.44/294.97 (0) 1160.44/294.97 Obligation: 1160.44/294.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1160.44/294.97 1160.44/294.97 1160.44/294.97 The TRS R consists of the following rules: 1160.44/294.97 1160.44/294.97 active(zeros) -> mark(cons(0, zeros)) 1160.44/294.97 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1160.44/294.97 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1160.44/294.97 active(U13(tt)) -> mark(tt) 1160.44/294.97 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1160.44/294.97 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1160.44/294.97 active(U23(tt)) -> mark(tt) 1160.44/294.97 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1160.44/294.97 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1160.44/294.97 active(U33(tt)) -> mark(tt) 1160.44/294.97 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1160.44/294.97 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1160.44/294.97 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1160.44/294.97 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1160.44/294.97 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1160.44/294.97 active(U46(tt)) -> mark(tt) 1160.44/294.97 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1160.44/294.97 active(U52(tt)) -> mark(tt) 1160.44/294.97 active(U61(tt)) -> mark(tt) 1160.44/294.97 active(U71(tt)) -> mark(tt) 1160.44/294.97 active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2)) 1160.44/294.97 active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2)) 1160.44/294.97 active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2)) 1160.44/294.97 active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2)) 1160.44/294.97 active(U85(tt, V2)) -> mark(U86(isNatList(V2))) 1160.44/294.97 active(U86(tt)) -> mark(tt) 1160.44/294.97 active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N)) 1160.46/294.97 active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N)) 1160.46/294.97 active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L)) 1160.46/294.97 active(U94(tt, L)) -> mark(s(length(L))) 1160.46/294.97 active(isNat(0)) -> mark(tt) 1160.46/294.97 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1160.46/294.97 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1160.46/294.97 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1160.46/294.97 active(isNatIList(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1160.46/294.97 active(isNatIListKind(nil)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1160.46/294.97 active(isNatKind(0)) -> mark(tt) 1160.46/294.97 active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1))) 1160.46/294.97 active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1))) 1160.46/294.97 active(isNatList(nil)) -> mark(tt) 1160.46/294.97 active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2)) 1160.46/294.97 active(length(nil)) -> mark(0) 1160.46/294.97 active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N)) 1160.46/294.97 active(cons(X1, X2)) -> cons(active(X1), X2) 1160.46/294.97 active(U11(X1, X2)) -> U11(active(X1), X2) 1160.46/294.97 active(U12(X1, X2)) -> U12(active(X1), X2) 1160.46/294.97 active(U13(X)) -> U13(active(X)) 1160.46/294.97 active(U21(X1, X2)) -> U21(active(X1), X2) 1160.46/294.97 active(U22(X1, X2)) -> U22(active(X1), X2) 1160.46/294.97 active(U23(X)) -> U23(active(X)) 1160.46/294.97 active(U31(X1, X2)) -> U31(active(X1), X2) 1160.46/294.97 active(U32(X1, X2)) -> U32(active(X1), X2) 1160.46/294.97 active(U33(X)) -> U33(active(X)) 1160.46/294.97 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1160.46/294.97 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1160.46/294.97 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1160.46/294.97 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1160.46/294.97 active(U45(X1, X2)) -> U45(active(X1), X2) 1160.46/294.97 active(U46(X)) -> U46(active(X)) 1160.46/294.97 active(U51(X1, X2)) -> U51(active(X1), X2) 1160.46/294.97 active(U52(X)) -> U52(active(X)) 1160.46/294.97 active(U61(X)) -> U61(active(X)) 1160.46/294.97 active(U71(X)) -> U71(active(X)) 1160.46/294.97 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 1160.46/294.97 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 1160.46/294.97 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 1160.46/294.97 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 1160.46/294.97 active(U85(X1, X2)) -> U85(active(X1), X2) 1160.46/294.97 active(U86(X)) -> U86(active(X)) 1160.46/294.97 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1160.46/294.97 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1160.46/294.97 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1160.46/294.97 active(U94(X1, X2)) -> U94(active(X1), X2) 1160.46/294.97 active(s(X)) -> s(active(X)) 1160.46/294.97 active(length(X)) -> length(active(X)) 1160.46/294.97 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1160.46/294.97 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1160.46/294.97 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1160.46/294.97 U13(mark(X)) -> mark(U13(X)) 1160.46/294.97 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1160.46/294.97 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1160.46/294.97 U23(mark(X)) -> mark(U23(X)) 1160.46/294.97 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1160.46/294.97 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1160.46/294.97 U33(mark(X)) -> mark(U33(X)) 1160.46/294.97 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1160.46/294.97 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1160.46/294.97 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1160.46/294.97 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1160.46/294.97 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1160.46/294.97 U46(mark(X)) -> mark(U46(X)) 1160.46/294.97 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1160.46/294.97 U52(mark(X)) -> mark(U52(X)) 1160.46/294.97 U61(mark(X)) -> mark(U61(X)) 1160.46/294.97 U71(mark(X)) -> mark(U71(X)) 1160.46/294.97 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 1160.46/294.97 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 1160.46/294.97 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 1160.46/294.97 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 1160.46/294.97 U85(mark(X1), X2) -> mark(U85(X1, X2)) 1160.46/294.97 U86(mark(X)) -> mark(U86(X)) 1160.46/294.97 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1160.46/294.97 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1160.46/294.97 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1160.46/294.97 U94(mark(X1), X2) -> mark(U94(X1, X2)) 1160.46/294.97 s(mark(X)) -> mark(s(X)) 1160.46/294.97 length(mark(X)) -> mark(length(X)) 1160.46/294.97 proper(zeros) -> ok(zeros) 1160.46/294.97 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1160.46/294.97 proper(0) -> ok(0) 1160.46/294.97 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1160.46/294.97 proper(tt) -> ok(tt) 1160.46/294.97 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1160.46/294.97 proper(U13(X)) -> U13(proper(X)) 1160.46/294.97 proper(isNatList(X)) -> isNatList(proper(X)) 1160.46/294.97 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1160.46/294.97 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatKind(X)) -> isNatKind(proper(X)) 1160.46/294.97 proper(U23(X)) -> U23(proper(X)) 1160.46/294.97 proper(isNat(X)) -> isNat(proper(X)) 1160.46/294.97 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1160.46/294.97 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1160.46/294.97 proper(U33(X)) -> U33(proper(X)) 1160.46/294.97 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1160.46/294.97 proper(U46(X)) -> U46(proper(X)) 1160.46/294.97 proper(isNatIList(X)) -> isNatIList(proper(X)) 1160.46/294.97 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1160.46/294.97 proper(U52(X)) -> U52(proper(X)) 1160.46/294.97 proper(U61(X)) -> U61(proper(X)) 1160.46/294.97 proper(U71(X)) -> U71(proper(X)) 1160.46/294.97 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U85(X1, X2)) -> U85(proper(X1), proper(X2)) 1160.46/294.97 proper(U86(X)) -> U86(proper(X)) 1160.46/294.97 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U94(X1, X2)) -> U94(proper(X1), proper(X2)) 1160.46/294.97 proper(s(X)) -> s(proper(X)) 1160.46/294.97 proper(length(X)) -> length(proper(X)) 1160.46/294.97 proper(nil) -> ok(nil) 1160.46/294.97 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1160.46/294.97 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1160.46/294.97 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1160.46/294.97 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1160.46/294.97 U13(ok(X)) -> ok(U13(X)) 1160.46/294.97 isNatList(ok(X)) -> ok(isNatList(X)) 1160.46/294.97 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1160.46/294.97 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1160.46/294.97 isNatKind(ok(X)) -> ok(isNatKind(X)) 1160.46/294.97 U23(ok(X)) -> ok(U23(X)) 1160.46/294.97 isNat(ok(X)) -> ok(isNat(X)) 1160.46/294.97 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1160.46/294.97 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1160.46/294.97 U33(ok(X)) -> ok(U33(X)) 1160.46/294.97 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1160.46/294.97 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1160.46/294.97 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1160.46/294.97 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1160.46/294.97 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1160.46/294.97 U46(ok(X)) -> ok(U46(X)) 1160.46/294.97 isNatIList(ok(X)) -> ok(isNatIList(X)) 1160.46/294.97 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1160.46/294.97 U52(ok(X)) -> ok(U52(X)) 1160.46/294.97 U61(ok(X)) -> ok(U61(X)) 1160.46/294.97 U71(ok(X)) -> ok(U71(X)) 1160.46/294.97 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 1160.46/294.97 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 1160.46/294.97 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 1160.46/294.97 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 1160.46/294.97 U85(ok(X1), ok(X2)) -> ok(U85(X1, X2)) 1160.46/294.97 U86(ok(X)) -> ok(U86(X)) 1160.46/294.97 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1160.46/294.97 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1160.46/294.97 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1160.46/294.97 U94(ok(X1), ok(X2)) -> ok(U94(X1, X2)) 1160.46/294.97 s(ok(X)) -> ok(s(X)) 1160.46/294.97 length(ok(X)) -> ok(length(X)) 1160.46/294.97 top(mark(X)) -> top(proper(X)) 1160.46/294.97 top(ok(X)) -> top(active(X)) 1160.46/294.97 1160.46/294.97 S is empty. 1160.46/294.97 Rewrite Strategy: FULL 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1160.46/294.97 Transformed a relative TRS into a decreasing-loop problem. 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (2) 1160.46/294.97 Obligation: 1160.46/294.97 Analyzing the following TRS for decreasing loops: 1160.46/294.97 1160.46/294.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1160.46/294.97 1160.46/294.97 1160.46/294.97 The TRS R consists of the following rules: 1160.46/294.97 1160.46/294.97 active(zeros) -> mark(cons(0, zeros)) 1160.46/294.97 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1160.46/294.97 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1160.46/294.97 active(U13(tt)) -> mark(tt) 1160.46/294.97 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1160.46/294.97 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1160.46/294.97 active(U23(tt)) -> mark(tt) 1160.46/294.97 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1160.46/294.97 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1160.46/294.97 active(U33(tt)) -> mark(tt) 1160.46/294.97 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1160.46/294.97 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1160.46/294.97 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1160.46/294.97 active(U46(tt)) -> mark(tt) 1160.46/294.97 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1160.46/294.97 active(U52(tt)) -> mark(tt) 1160.46/294.97 active(U61(tt)) -> mark(tt) 1160.46/294.97 active(U71(tt)) -> mark(tt) 1160.46/294.97 active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2)) 1160.46/294.97 active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2)) 1160.46/294.97 active(U85(tt, V2)) -> mark(U86(isNatList(V2))) 1160.46/294.97 active(U86(tt)) -> mark(tt) 1160.46/294.97 active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N)) 1160.46/294.97 active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N)) 1160.46/294.97 active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L)) 1160.46/294.97 active(U94(tt, L)) -> mark(s(length(L))) 1160.46/294.97 active(isNat(0)) -> mark(tt) 1160.46/294.97 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1160.46/294.97 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1160.46/294.97 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1160.46/294.97 active(isNatIList(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1160.46/294.97 active(isNatIListKind(nil)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1160.46/294.97 active(isNatKind(0)) -> mark(tt) 1160.46/294.97 active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1))) 1160.46/294.97 active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1))) 1160.46/294.97 active(isNatList(nil)) -> mark(tt) 1160.46/294.97 active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2)) 1160.46/294.97 active(length(nil)) -> mark(0) 1160.46/294.97 active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N)) 1160.46/294.97 active(cons(X1, X2)) -> cons(active(X1), X2) 1160.46/294.97 active(U11(X1, X2)) -> U11(active(X1), X2) 1160.46/294.97 active(U12(X1, X2)) -> U12(active(X1), X2) 1160.46/294.97 active(U13(X)) -> U13(active(X)) 1160.46/294.97 active(U21(X1, X2)) -> U21(active(X1), X2) 1160.46/294.97 active(U22(X1, X2)) -> U22(active(X1), X2) 1160.46/294.97 active(U23(X)) -> U23(active(X)) 1160.46/294.97 active(U31(X1, X2)) -> U31(active(X1), X2) 1160.46/294.97 active(U32(X1, X2)) -> U32(active(X1), X2) 1160.46/294.97 active(U33(X)) -> U33(active(X)) 1160.46/294.97 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1160.46/294.97 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1160.46/294.97 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1160.46/294.97 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1160.46/294.97 active(U45(X1, X2)) -> U45(active(X1), X2) 1160.46/294.97 active(U46(X)) -> U46(active(X)) 1160.46/294.97 active(U51(X1, X2)) -> U51(active(X1), X2) 1160.46/294.97 active(U52(X)) -> U52(active(X)) 1160.46/294.97 active(U61(X)) -> U61(active(X)) 1160.46/294.97 active(U71(X)) -> U71(active(X)) 1160.46/294.97 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 1160.46/294.97 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 1160.46/294.97 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 1160.46/294.97 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 1160.46/294.97 active(U85(X1, X2)) -> U85(active(X1), X2) 1160.46/294.97 active(U86(X)) -> U86(active(X)) 1160.46/294.97 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1160.46/294.97 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1160.46/294.97 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1160.46/294.97 active(U94(X1, X2)) -> U94(active(X1), X2) 1160.46/294.97 active(s(X)) -> s(active(X)) 1160.46/294.97 active(length(X)) -> length(active(X)) 1160.46/294.97 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1160.46/294.97 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1160.46/294.97 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1160.46/294.97 U13(mark(X)) -> mark(U13(X)) 1160.46/294.97 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1160.46/294.97 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1160.46/294.97 U23(mark(X)) -> mark(U23(X)) 1160.46/294.97 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1160.46/294.97 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1160.46/294.97 U33(mark(X)) -> mark(U33(X)) 1160.46/294.97 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1160.46/294.97 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1160.46/294.97 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1160.46/294.97 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1160.46/294.97 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1160.46/294.97 U46(mark(X)) -> mark(U46(X)) 1160.46/294.97 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1160.46/294.97 U52(mark(X)) -> mark(U52(X)) 1160.46/294.97 U61(mark(X)) -> mark(U61(X)) 1160.46/294.97 U71(mark(X)) -> mark(U71(X)) 1160.46/294.97 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 1160.46/294.97 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 1160.46/294.97 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 1160.46/294.97 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 1160.46/294.97 U85(mark(X1), X2) -> mark(U85(X1, X2)) 1160.46/294.97 U86(mark(X)) -> mark(U86(X)) 1160.46/294.97 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1160.46/294.97 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1160.46/294.97 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1160.46/294.97 U94(mark(X1), X2) -> mark(U94(X1, X2)) 1160.46/294.97 s(mark(X)) -> mark(s(X)) 1160.46/294.97 length(mark(X)) -> mark(length(X)) 1160.46/294.97 proper(zeros) -> ok(zeros) 1160.46/294.97 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1160.46/294.97 proper(0) -> ok(0) 1160.46/294.97 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1160.46/294.97 proper(tt) -> ok(tt) 1160.46/294.97 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1160.46/294.97 proper(U13(X)) -> U13(proper(X)) 1160.46/294.97 proper(isNatList(X)) -> isNatList(proper(X)) 1160.46/294.97 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1160.46/294.97 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatKind(X)) -> isNatKind(proper(X)) 1160.46/294.97 proper(U23(X)) -> U23(proper(X)) 1160.46/294.97 proper(isNat(X)) -> isNat(proper(X)) 1160.46/294.97 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1160.46/294.97 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1160.46/294.97 proper(U33(X)) -> U33(proper(X)) 1160.46/294.97 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1160.46/294.97 proper(U46(X)) -> U46(proper(X)) 1160.46/294.97 proper(isNatIList(X)) -> isNatIList(proper(X)) 1160.46/294.97 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1160.46/294.97 proper(U52(X)) -> U52(proper(X)) 1160.46/294.97 proper(U61(X)) -> U61(proper(X)) 1160.46/294.97 proper(U71(X)) -> U71(proper(X)) 1160.46/294.97 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U85(X1, X2)) -> U85(proper(X1), proper(X2)) 1160.46/294.97 proper(U86(X)) -> U86(proper(X)) 1160.46/294.97 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U94(X1, X2)) -> U94(proper(X1), proper(X2)) 1160.46/294.97 proper(s(X)) -> s(proper(X)) 1160.46/294.97 proper(length(X)) -> length(proper(X)) 1160.46/294.97 proper(nil) -> ok(nil) 1160.46/294.97 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1160.46/294.97 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1160.46/294.97 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1160.46/294.97 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1160.46/294.97 U13(ok(X)) -> ok(U13(X)) 1160.46/294.97 isNatList(ok(X)) -> ok(isNatList(X)) 1160.46/294.97 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1160.46/294.97 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1160.46/294.97 isNatKind(ok(X)) -> ok(isNatKind(X)) 1160.46/294.97 U23(ok(X)) -> ok(U23(X)) 1160.46/294.97 isNat(ok(X)) -> ok(isNat(X)) 1160.46/294.97 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1160.46/294.97 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1160.46/294.97 U33(ok(X)) -> ok(U33(X)) 1160.46/294.97 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1160.46/294.97 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1160.46/294.97 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1160.46/294.97 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1160.46/294.97 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1160.46/294.97 U46(ok(X)) -> ok(U46(X)) 1160.46/294.97 isNatIList(ok(X)) -> ok(isNatIList(X)) 1160.46/294.97 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1160.46/294.97 U52(ok(X)) -> ok(U52(X)) 1160.46/294.97 U61(ok(X)) -> ok(U61(X)) 1160.46/294.97 U71(ok(X)) -> ok(U71(X)) 1160.46/294.97 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 1160.46/294.97 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 1160.46/294.97 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 1160.46/294.97 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 1160.46/294.97 U85(ok(X1), ok(X2)) -> ok(U85(X1, X2)) 1160.46/294.97 U86(ok(X)) -> ok(U86(X)) 1160.46/294.97 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1160.46/294.97 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1160.46/294.97 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1160.46/294.97 U94(ok(X1), ok(X2)) -> ok(U94(X1, X2)) 1160.46/294.97 s(ok(X)) -> ok(s(X)) 1160.46/294.97 length(ok(X)) -> ok(length(X)) 1160.46/294.97 top(mark(X)) -> top(proper(X)) 1160.46/294.97 top(ok(X)) -> top(active(X)) 1160.46/294.97 1160.46/294.97 S is empty. 1160.46/294.97 Rewrite Strategy: FULL 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1160.46/294.97 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1160.46/294.97 1160.46/294.97 The rewrite sequence 1160.46/294.97 1160.46/294.97 U23(mark(X)) ->^+ mark(U23(X)) 1160.46/294.97 1160.46/294.97 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1160.46/294.97 1160.46/294.97 The pumping substitution is [X / mark(X)]. 1160.46/294.97 1160.46/294.97 The result substitution is [ ]. 1160.46/294.97 1160.46/294.97 1160.46/294.97 1160.46/294.97 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (4) 1160.46/294.97 Complex Obligation (BEST) 1160.46/294.97 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (5) 1160.46/294.97 Obligation: 1160.46/294.97 Proved the lower bound n^1 for the following obligation: 1160.46/294.97 1160.46/294.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1160.46/294.97 1160.46/294.97 1160.46/294.97 The TRS R consists of the following rules: 1160.46/294.97 1160.46/294.97 active(zeros) -> mark(cons(0, zeros)) 1160.46/294.97 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1160.46/294.97 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1160.46/294.97 active(U13(tt)) -> mark(tt) 1160.46/294.97 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1160.46/294.97 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1160.46/294.97 active(U23(tt)) -> mark(tt) 1160.46/294.97 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1160.46/294.97 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1160.46/294.97 active(U33(tt)) -> mark(tt) 1160.46/294.97 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1160.46/294.97 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1160.46/294.97 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1160.46/294.97 active(U46(tt)) -> mark(tt) 1160.46/294.97 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1160.46/294.97 active(U52(tt)) -> mark(tt) 1160.46/294.97 active(U61(tt)) -> mark(tt) 1160.46/294.97 active(U71(tt)) -> mark(tt) 1160.46/294.97 active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2)) 1160.46/294.97 active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2)) 1160.46/294.97 active(U85(tt, V2)) -> mark(U86(isNatList(V2))) 1160.46/294.97 active(U86(tt)) -> mark(tt) 1160.46/294.97 active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N)) 1160.46/294.97 active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N)) 1160.46/294.97 active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L)) 1160.46/294.97 active(U94(tt, L)) -> mark(s(length(L))) 1160.46/294.97 active(isNat(0)) -> mark(tt) 1160.46/294.97 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1160.46/294.97 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1160.46/294.97 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1160.46/294.97 active(isNatIList(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1160.46/294.97 active(isNatIListKind(nil)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1160.46/294.97 active(isNatKind(0)) -> mark(tt) 1160.46/294.97 active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1))) 1160.46/294.97 active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1))) 1160.46/294.97 active(isNatList(nil)) -> mark(tt) 1160.46/294.97 active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2)) 1160.46/294.97 active(length(nil)) -> mark(0) 1160.46/294.97 active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N)) 1160.46/294.97 active(cons(X1, X2)) -> cons(active(X1), X2) 1160.46/294.97 active(U11(X1, X2)) -> U11(active(X1), X2) 1160.46/294.97 active(U12(X1, X2)) -> U12(active(X1), X2) 1160.46/294.97 active(U13(X)) -> U13(active(X)) 1160.46/294.97 active(U21(X1, X2)) -> U21(active(X1), X2) 1160.46/294.97 active(U22(X1, X2)) -> U22(active(X1), X2) 1160.46/294.97 active(U23(X)) -> U23(active(X)) 1160.46/294.97 active(U31(X1, X2)) -> U31(active(X1), X2) 1160.46/294.97 active(U32(X1, X2)) -> U32(active(X1), X2) 1160.46/294.97 active(U33(X)) -> U33(active(X)) 1160.46/294.97 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1160.46/294.97 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1160.46/294.97 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1160.46/294.97 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1160.46/294.97 active(U45(X1, X2)) -> U45(active(X1), X2) 1160.46/294.97 active(U46(X)) -> U46(active(X)) 1160.46/294.97 active(U51(X1, X2)) -> U51(active(X1), X2) 1160.46/294.97 active(U52(X)) -> U52(active(X)) 1160.46/294.97 active(U61(X)) -> U61(active(X)) 1160.46/294.97 active(U71(X)) -> U71(active(X)) 1160.46/294.97 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 1160.46/294.97 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 1160.46/294.97 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 1160.46/294.97 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 1160.46/294.97 active(U85(X1, X2)) -> U85(active(X1), X2) 1160.46/294.97 active(U86(X)) -> U86(active(X)) 1160.46/294.97 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1160.46/294.97 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1160.46/294.97 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1160.46/294.97 active(U94(X1, X2)) -> U94(active(X1), X2) 1160.46/294.97 active(s(X)) -> s(active(X)) 1160.46/294.97 active(length(X)) -> length(active(X)) 1160.46/294.97 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1160.46/294.97 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1160.46/294.97 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1160.46/294.97 U13(mark(X)) -> mark(U13(X)) 1160.46/294.97 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1160.46/294.97 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1160.46/294.97 U23(mark(X)) -> mark(U23(X)) 1160.46/294.97 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1160.46/294.97 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1160.46/294.97 U33(mark(X)) -> mark(U33(X)) 1160.46/294.97 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1160.46/294.97 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1160.46/294.97 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1160.46/294.97 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1160.46/294.97 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1160.46/294.97 U46(mark(X)) -> mark(U46(X)) 1160.46/294.97 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1160.46/294.97 U52(mark(X)) -> mark(U52(X)) 1160.46/294.97 U61(mark(X)) -> mark(U61(X)) 1160.46/294.97 U71(mark(X)) -> mark(U71(X)) 1160.46/294.97 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 1160.46/294.97 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 1160.46/294.97 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 1160.46/294.97 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 1160.46/294.97 U85(mark(X1), X2) -> mark(U85(X1, X2)) 1160.46/294.97 U86(mark(X)) -> mark(U86(X)) 1160.46/294.97 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1160.46/294.97 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1160.46/294.97 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1160.46/294.97 U94(mark(X1), X2) -> mark(U94(X1, X2)) 1160.46/294.97 s(mark(X)) -> mark(s(X)) 1160.46/294.97 length(mark(X)) -> mark(length(X)) 1160.46/294.97 proper(zeros) -> ok(zeros) 1160.46/294.97 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1160.46/294.97 proper(0) -> ok(0) 1160.46/294.97 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1160.46/294.97 proper(tt) -> ok(tt) 1160.46/294.97 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1160.46/294.97 proper(U13(X)) -> U13(proper(X)) 1160.46/294.97 proper(isNatList(X)) -> isNatList(proper(X)) 1160.46/294.97 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1160.46/294.97 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatKind(X)) -> isNatKind(proper(X)) 1160.46/294.97 proper(U23(X)) -> U23(proper(X)) 1160.46/294.97 proper(isNat(X)) -> isNat(proper(X)) 1160.46/294.97 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1160.46/294.97 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1160.46/294.97 proper(U33(X)) -> U33(proper(X)) 1160.46/294.97 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1160.46/294.97 proper(U46(X)) -> U46(proper(X)) 1160.46/294.97 proper(isNatIList(X)) -> isNatIList(proper(X)) 1160.46/294.97 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1160.46/294.97 proper(U52(X)) -> U52(proper(X)) 1160.46/294.97 proper(U61(X)) -> U61(proper(X)) 1160.46/294.97 proper(U71(X)) -> U71(proper(X)) 1160.46/294.97 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U85(X1, X2)) -> U85(proper(X1), proper(X2)) 1160.46/294.97 proper(U86(X)) -> U86(proper(X)) 1160.46/294.97 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U94(X1, X2)) -> U94(proper(X1), proper(X2)) 1160.46/294.97 proper(s(X)) -> s(proper(X)) 1160.46/294.97 proper(length(X)) -> length(proper(X)) 1160.46/294.97 proper(nil) -> ok(nil) 1160.46/294.97 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1160.46/294.97 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1160.46/294.97 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1160.46/294.97 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1160.46/294.97 U13(ok(X)) -> ok(U13(X)) 1160.46/294.97 isNatList(ok(X)) -> ok(isNatList(X)) 1160.46/294.97 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1160.46/294.97 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1160.46/294.97 isNatKind(ok(X)) -> ok(isNatKind(X)) 1160.46/294.97 U23(ok(X)) -> ok(U23(X)) 1160.46/294.97 isNat(ok(X)) -> ok(isNat(X)) 1160.46/294.97 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1160.46/294.97 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1160.46/294.97 U33(ok(X)) -> ok(U33(X)) 1160.46/294.97 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1160.46/294.97 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1160.46/294.97 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1160.46/294.97 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1160.46/294.97 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1160.46/294.97 U46(ok(X)) -> ok(U46(X)) 1160.46/294.97 isNatIList(ok(X)) -> ok(isNatIList(X)) 1160.46/294.97 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1160.46/294.97 U52(ok(X)) -> ok(U52(X)) 1160.46/294.97 U61(ok(X)) -> ok(U61(X)) 1160.46/294.97 U71(ok(X)) -> ok(U71(X)) 1160.46/294.97 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 1160.46/294.97 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 1160.46/294.97 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 1160.46/294.97 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 1160.46/294.97 U85(ok(X1), ok(X2)) -> ok(U85(X1, X2)) 1160.46/294.97 U86(ok(X)) -> ok(U86(X)) 1160.46/294.97 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1160.46/294.97 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1160.46/294.97 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1160.46/294.97 U94(ok(X1), ok(X2)) -> ok(U94(X1, X2)) 1160.46/294.97 s(ok(X)) -> ok(s(X)) 1160.46/294.97 length(ok(X)) -> ok(length(X)) 1160.46/294.97 top(mark(X)) -> top(proper(X)) 1160.46/294.97 top(ok(X)) -> top(active(X)) 1160.46/294.97 1160.46/294.97 S is empty. 1160.46/294.97 Rewrite Strategy: FULL 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (6) LowerBoundPropagationProof (FINISHED) 1160.46/294.97 Propagated lower bound. 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (7) 1160.46/294.97 BOUNDS(n^1, INF) 1160.46/294.97 1160.46/294.97 ---------------------------------------- 1160.46/294.97 1160.46/294.97 (8) 1160.46/294.97 Obligation: 1160.46/294.97 Analyzing the following TRS for decreasing loops: 1160.46/294.97 1160.46/294.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1160.46/294.97 1160.46/294.97 1160.46/294.97 The TRS R consists of the following rules: 1160.46/294.97 1160.46/294.97 active(zeros) -> mark(cons(0, zeros)) 1160.46/294.97 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1160.46/294.97 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1160.46/294.97 active(U13(tt)) -> mark(tt) 1160.46/294.97 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1160.46/294.97 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1160.46/294.97 active(U23(tt)) -> mark(tt) 1160.46/294.97 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1160.46/294.97 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1160.46/294.97 active(U33(tt)) -> mark(tt) 1160.46/294.97 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1160.46/294.97 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1160.46/294.97 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1160.46/294.97 active(U46(tt)) -> mark(tt) 1160.46/294.97 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1160.46/294.97 active(U52(tt)) -> mark(tt) 1160.46/294.97 active(U61(tt)) -> mark(tt) 1160.46/294.97 active(U71(tt)) -> mark(tt) 1160.46/294.97 active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2)) 1160.46/294.97 active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2)) 1160.46/294.97 active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2)) 1160.46/294.97 active(U85(tt, V2)) -> mark(U86(isNatList(V2))) 1160.46/294.97 active(U86(tt)) -> mark(tt) 1160.46/294.97 active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N)) 1160.46/294.97 active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N)) 1160.46/294.97 active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L)) 1160.46/294.97 active(U94(tt, L)) -> mark(s(length(L))) 1160.46/294.97 active(isNat(0)) -> mark(tt) 1160.46/294.97 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1160.46/294.97 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1160.46/294.97 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1160.46/294.97 active(isNatIList(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1160.46/294.97 active(isNatIListKind(nil)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(zeros)) -> mark(tt) 1160.46/294.97 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1160.46/294.97 active(isNatKind(0)) -> mark(tt) 1160.46/294.97 active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1))) 1160.46/294.97 active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1))) 1160.46/294.97 active(isNatList(nil)) -> mark(tt) 1160.46/294.97 active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2)) 1160.46/294.97 active(length(nil)) -> mark(0) 1160.46/294.97 active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N)) 1160.46/294.97 active(cons(X1, X2)) -> cons(active(X1), X2) 1160.46/294.97 active(U11(X1, X2)) -> U11(active(X1), X2) 1160.46/294.97 active(U12(X1, X2)) -> U12(active(X1), X2) 1160.46/294.97 active(U13(X)) -> U13(active(X)) 1160.46/294.97 active(U21(X1, X2)) -> U21(active(X1), X2) 1160.46/294.97 active(U22(X1, X2)) -> U22(active(X1), X2) 1160.46/294.97 active(U23(X)) -> U23(active(X)) 1160.46/294.97 active(U31(X1, X2)) -> U31(active(X1), X2) 1160.46/294.97 active(U32(X1, X2)) -> U32(active(X1), X2) 1160.46/294.97 active(U33(X)) -> U33(active(X)) 1160.46/294.97 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1160.46/294.97 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1160.46/294.97 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1160.46/294.97 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1160.46/294.97 active(U45(X1, X2)) -> U45(active(X1), X2) 1160.46/294.97 active(U46(X)) -> U46(active(X)) 1160.46/294.97 active(U51(X1, X2)) -> U51(active(X1), X2) 1160.46/294.97 active(U52(X)) -> U52(active(X)) 1160.46/294.97 active(U61(X)) -> U61(active(X)) 1160.46/294.97 active(U71(X)) -> U71(active(X)) 1160.46/294.97 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 1160.46/294.97 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 1160.46/294.97 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 1160.46/294.97 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 1160.46/294.97 active(U85(X1, X2)) -> U85(active(X1), X2) 1160.46/294.97 active(U86(X)) -> U86(active(X)) 1160.46/294.97 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1160.46/294.97 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1160.46/294.97 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1160.46/294.97 active(U94(X1, X2)) -> U94(active(X1), X2) 1160.46/294.97 active(s(X)) -> s(active(X)) 1160.46/294.97 active(length(X)) -> length(active(X)) 1160.46/294.97 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1160.46/294.97 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1160.46/294.97 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1160.46/294.97 U13(mark(X)) -> mark(U13(X)) 1160.46/294.97 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1160.46/294.97 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1160.46/294.97 U23(mark(X)) -> mark(U23(X)) 1160.46/294.97 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1160.46/294.97 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1160.46/294.97 U33(mark(X)) -> mark(U33(X)) 1160.46/294.97 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1160.46/294.97 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1160.46/294.97 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1160.46/294.97 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1160.46/294.97 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1160.46/294.97 U46(mark(X)) -> mark(U46(X)) 1160.46/294.97 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1160.46/294.97 U52(mark(X)) -> mark(U52(X)) 1160.46/294.97 U61(mark(X)) -> mark(U61(X)) 1160.46/294.97 U71(mark(X)) -> mark(U71(X)) 1160.46/294.97 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 1160.46/294.97 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 1160.46/294.97 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 1160.46/294.97 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 1160.46/294.97 U85(mark(X1), X2) -> mark(U85(X1, X2)) 1160.46/294.97 U86(mark(X)) -> mark(U86(X)) 1160.46/294.97 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1160.46/294.97 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1160.46/294.97 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1160.46/294.97 U94(mark(X1), X2) -> mark(U94(X1, X2)) 1160.46/294.97 s(mark(X)) -> mark(s(X)) 1160.46/294.97 length(mark(X)) -> mark(length(X)) 1160.46/294.97 proper(zeros) -> ok(zeros) 1160.46/294.97 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1160.46/294.97 proper(0) -> ok(0) 1160.46/294.97 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1160.46/294.97 proper(tt) -> ok(tt) 1160.46/294.97 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1160.46/294.97 proper(U13(X)) -> U13(proper(X)) 1160.46/294.97 proper(isNatList(X)) -> isNatList(proper(X)) 1160.46/294.97 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1160.46/294.97 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1160.46/294.97 proper(isNatKind(X)) -> isNatKind(proper(X)) 1160.46/294.97 proper(U23(X)) -> U23(proper(X)) 1160.46/294.97 proper(isNat(X)) -> isNat(proper(X)) 1160.46/294.97 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1160.46/294.97 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1160.46/294.97 proper(U33(X)) -> U33(proper(X)) 1160.46/294.97 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1160.46/294.97 proper(U46(X)) -> U46(proper(X)) 1160.46/294.97 proper(isNatIList(X)) -> isNatIList(proper(X)) 1160.46/294.97 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1160.46/294.97 proper(U52(X)) -> U52(proper(X)) 1160.46/294.97 proper(U61(X)) -> U61(proper(X)) 1160.46/294.97 proper(U71(X)) -> U71(proper(X)) 1160.46/294.97 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U85(X1, X2)) -> U85(proper(X1), proper(X2)) 1160.46/294.97 proper(U86(X)) -> U86(proper(X)) 1160.46/294.97 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1160.46/294.97 proper(U94(X1, X2)) -> U94(proper(X1), proper(X2)) 1160.46/294.97 proper(s(X)) -> s(proper(X)) 1160.46/294.97 proper(length(X)) -> length(proper(X)) 1160.46/294.97 proper(nil) -> ok(nil) 1160.46/294.97 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1160.46/294.97 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1160.46/294.97 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1160.46/294.97 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1160.46/294.97 U13(ok(X)) -> ok(U13(X)) 1160.46/294.97 isNatList(ok(X)) -> ok(isNatList(X)) 1160.46/294.97 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1160.46/294.97 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1160.46/294.97 isNatKind(ok(X)) -> ok(isNatKind(X)) 1160.46/294.97 U23(ok(X)) -> ok(U23(X)) 1160.46/294.97 isNat(ok(X)) -> ok(isNat(X)) 1160.46/294.97 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1160.46/294.97 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1160.46/294.97 U33(ok(X)) -> ok(U33(X)) 1160.46/294.97 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1160.46/294.97 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1160.46/294.97 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1160.46/294.97 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1160.46/294.97 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1160.46/294.97 U46(ok(X)) -> ok(U46(X)) 1160.46/294.97 isNatIList(ok(X)) -> ok(isNatIList(X)) 1160.46/294.97 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1160.46/294.97 U52(ok(X)) -> ok(U52(X)) 1160.46/294.97 U61(ok(X)) -> ok(U61(X)) 1160.46/294.97 U71(ok(X)) -> ok(U71(X)) 1160.46/294.97 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 1160.46/294.97 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 1160.46/294.97 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 1160.46/294.97 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 1160.46/294.97 U85(ok(X1), ok(X2)) -> ok(U85(X1, X2)) 1160.46/294.97 U86(ok(X)) -> ok(U86(X)) 1160.46/294.97 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1160.46/294.97 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1160.46/294.97 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1160.46/294.97 U94(ok(X1), ok(X2)) -> ok(U94(X1, X2)) 1160.46/294.97 s(ok(X)) -> ok(s(X)) 1160.46/294.97 length(ok(X)) -> ok(length(X)) 1160.46/294.97 top(mark(X)) -> top(proper(X)) 1160.46/294.97 top(ok(X)) -> top(active(X)) 1160.46/294.97 1160.46/294.97 S is empty. 1160.46/294.97 Rewrite Strategy: FULL 1160.55/295.04 EOF