1143.39/291.61 WORST_CASE(Omega(n^1), ?) 1158.63/295.45 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1158.63/295.45 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1158.63/295.45 1158.63/295.45 1158.63/295.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.63/295.45 1158.63/295.45 (0) CpxTRS 1158.63/295.45 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1158.63/295.45 (2) TRS for Loop Detection 1158.63/295.45 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1158.63/295.45 (4) BEST 1158.63/295.45 (5) proven lower bound 1158.63/295.45 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1158.63/295.45 (7) BOUNDS(n^1, INF) 1158.63/295.45 (8) TRS for Loop Detection 1158.63/295.45 1158.63/295.45 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (0) 1158.63/295.45 Obligation: 1158.63/295.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.63/295.45 1158.63/295.45 1158.63/295.45 The TRS R consists of the following rules: 1158.63/295.45 1158.63/295.45 active(zeros) -> mark(cons(0, zeros)) 1158.63/295.45 active(U11(tt, V1)) -> mark(U12(isNatList(V1))) 1158.63/295.45 active(U12(tt)) -> mark(tt) 1158.63/295.45 active(U21(tt, V1)) -> mark(U22(isNat(V1))) 1158.63/295.45 active(U22(tt)) -> mark(tt) 1158.63/295.45 active(U31(tt, V)) -> mark(U32(isNatList(V))) 1158.63/295.45 active(U32(tt)) -> mark(tt) 1158.63/295.45 active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) 1158.63/295.45 active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) 1158.63/295.45 active(U43(tt)) -> mark(tt) 1158.63/295.45 active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) 1158.63/295.45 active(U52(tt, V2)) -> mark(U53(isNatList(V2))) 1158.63/295.45 active(U53(tt)) -> mark(tt) 1158.63/295.45 active(U61(tt, L)) -> mark(s(length(L))) 1158.63/295.45 active(and(tt, X)) -> mark(X) 1158.63/295.45 active(isNat(0)) -> mark(tt) 1158.63/295.45 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1158.63/295.45 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1158.63/295.45 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1158.63/295.45 active(isNatIList(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(isNatIListKind(nil)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) 1158.63/295.45 active(isNatKind(0)) -> mark(tt) 1158.63/295.45 active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) 1158.63/295.45 active(isNatKind(s(V1))) -> mark(isNatKind(V1)) 1158.63/295.45 active(isNatList(nil)) -> mark(tt) 1158.63/295.45 active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(length(nil)) -> mark(0) 1158.63/295.45 active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) 1158.63/295.45 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.63/295.45 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.63/295.45 active(U12(X)) -> U12(active(X)) 1158.63/295.45 active(U21(X1, X2)) -> U21(active(X1), X2) 1158.63/295.45 active(U22(X)) -> U22(active(X)) 1158.63/295.45 active(U31(X1, X2)) -> U31(active(X1), X2) 1158.63/295.45 active(U32(X)) -> U32(active(X)) 1158.63/295.45 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1158.63/295.45 active(U42(X1, X2)) -> U42(active(X1), X2) 1158.63/295.45 active(U43(X)) -> U43(active(X)) 1158.63/295.45 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 1158.63/295.45 active(U52(X1, X2)) -> U52(active(X1), X2) 1158.63/295.45 active(U53(X)) -> U53(active(X)) 1158.63/295.45 active(U61(X1, X2)) -> U61(active(X1), X2) 1158.63/295.45 active(s(X)) -> s(active(X)) 1158.63/295.45 active(length(X)) -> length(active(X)) 1158.63/295.45 active(and(X1, X2)) -> and(active(X1), X2) 1158.63/295.45 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.63/295.45 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.63/295.45 U12(mark(X)) -> mark(U12(X)) 1158.63/295.45 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1158.63/295.45 U22(mark(X)) -> mark(U22(X)) 1158.63/295.45 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1158.63/295.45 U32(mark(X)) -> mark(U32(X)) 1158.63/295.45 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1158.63/295.45 U42(mark(X1), X2) -> mark(U42(X1, X2)) 1158.63/295.45 U43(mark(X)) -> mark(U43(X)) 1158.63/295.45 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 1158.63/295.45 U52(mark(X1), X2) -> mark(U52(X1, X2)) 1158.63/295.45 U53(mark(X)) -> mark(U53(X)) 1158.63/295.45 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1158.63/295.45 s(mark(X)) -> mark(s(X)) 1158.63/295.45 length(mark(X)) -> mark(length(X)) 1158.63/295.45 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.63/295.45 proper(zeros) -> ok(zeros) 1158.63/295.45 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.63/295.45 proper(0) -> ok(0) 1158.63/295.45 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.63/295.45 proper(tt) -> ok(tt) 1158.63/295.45 proper(U12(X)) -> U12(proper(X)) 1158.63/295.45 proper(isNatList(X)) -> isNatList(proper(X)) 1158.63/295.45 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1158.63/295.45 proper(U22(X)) -> U22(proper(X)) 1158.63/295.45 proper(isNat(X)) -> isNat(proper(X)) 1158.63/295.45 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1158.63/295.45 proper(U32(X)) -> U32(proper(X)) 1158.63/295.45 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 1158.63/295.45 proper(U43(X)) -> U43(proper(X)) 1158.63/295.45 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.63/295.45 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 1158.63/295.45 proper(U53(X)) -> U53(proper(X)) 1158.63/295.45 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1158.63/295.45 proper(s(X)) -> s(proper(X)) 1158.63/295.45 proper(length(X)) -> length(proper(X)) 1158.63/295.45 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.63/295.45 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1158.63/295.45 proper(isNatKind(X)) -> isNatKind(proper(X)) 1158.63/295.45 proper(nil) -> ok(nil) 1158.63/295.45 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.63/295.45 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.63/295.45 U12(ok(X)) -> ok(U12(X)) 1158.63/295.45 isNatList(ok(X)) -> ok(isNatList(X)) 1158.63/295.45 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1158.63/295.45 U22(ok(X)) -> ok(U22(X)) 1158.63/295.45 isNat(ok(X)) -> ok(isNat(X)) 1158.63/295.45 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1158.63/295.45 U32(ok(X)) -> ok(U32(X)) 1158.63/295.45 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1158.63/295.45 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 1158.63/295.45 U43(ok(X)) -> ok(U43(X)) 1158.63/295.45 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.63/295.45 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 1158.63/295.45 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 1158.63/295.45 U53(ok(X)) -> ok(U53(X)) 1158.63/295.45 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1158.63/295.45 s(ok(X)) -> ok(s(X)) 1158.63/295.45 length(ok(X)) -> ok(length(X)) 1158.63/295.45 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.63/295.45 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1158.63/295.45 isNatKind(ok(X)) -> ok(isNatKind(X)) 1158.63/295.45 top(mark(X)) -> top(proper(X)) 1158.63/295.45 top(ok(X)) -> top(active(X)) 1158.63/295.45 1158.63/295.45 S is empty. 1158.63/295.45 Rewrite Strategy: FULL 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1158.63/295.45 Transformed a relative TRS into a decreasing-loop problem. 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (2) 1158.63/295.45 Obligation: 1158.63/295.45 Analyzing the following TRS for decreasing loops: 1158.63/295.45 1158.63/295.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.63/295.45 1158.63/295.45 1158.63/295.45 The TRS R consists of the following rules: 1158.63/295.45 1158.63/295.45 active(zeros) -> mark(cons(0, zeros)) 1158.63/295.45 active(U11(tt, V1)) -> mark(U12(isNatList(V1))) 1158.63/295.45 active(U12(tt)) -> mark(tt) 1158.63/295.45 active(U21(tt, V1)) -> mark(U22(isNat(V1))) 1158.63/295.45 active(U22(tt)) -> mark(tt) 1158.63/295.45 active(U31(tt, V)) -> mark(U32(isNatList(V))) 1158.63/295.45 active(U32(tt)) -> mark(tt) 1158.63/295.45 active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) 1158.63/295.45 active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) 1158.63/295.45 active(U43(tt)) -> mark(tt) 1158.63/295.45 active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) 1158.63/295.45 active(U52(tt, V2)) -> mark(U53(isNatList(V2))) 1158.63/295.45 active(U53(tt)) -> mark(tt) 1158.63/295.45 active(U61(tt, L)) -> mark(s(length(L))) 1158.63/295.45 active(and(tt, X)) -> mark(X) 1158.63/295.45 active(isNat(0)) -> mark(tt) 1158.63/295.45 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1158.63/295.45 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1158.63/295.45 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1158.63/295.45 active(isNatIList(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(isNatIListKind(nil)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) 1158.63/295.45 active(isNatKind(0)) -> mark(tt) 1158.63/295.45 active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) 1158.63/295.45 active(isNatKind(s(V1))) -> mark(isNatKind(V1)) 1158.63/295.45 active(isNatList(nil)) -> mark(tt) 1158.63/295.45 active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(length(nil)) -> mark(0) 1158.63/295.45 active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) 1158.63/295.45 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.63/295.45 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.63/295.45 active(U12(X)) -> U12(active(X)) 1158.63/295.45 active(U21(X1, X2)) -> U21(active(X1), X2) 1158.63/295.45 active(U22(X)) -> U22(active(X)) 1158.63/295.45 active(U31(X1, X2)) -> U31(active(X1), X2) 1158.63/295.45 active(U32(X)) -> U32(active(X)) 1158.63/295.45 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1158.63/295.45 active(U42(X1, X2)) -> U42(active(X1), X2) 1158.63/295.45 active(U43(X)) -> U43(active(X)) 1158.63/295.45 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 1158.63/295.45 active(U52(X1, X2)) -> U52(active(X1), X2) 1158.63/295.45 active(U53(X)) -> U53(active(X)) 1158.63/295.45 active(U61(X1, X2)) -> U61(active(X1), X2) 1158.63/295.45 active(s(X)) -> s(active(X)) 1158.63/295.45 active(length(X)) -> length(active(X)) 1158.63/295.45 active(and(X1, X2)) -> and(active(X1), X2) 1158.63/295.45 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.63/295.45 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.63/295.45 U12(mark(X)) -> mark(U12(X)) 1158.63/295.45 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1158.63/295.45 U22(mark(X)) -> mark(U22(X)) 1158.63/295.45 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1158.63/295.45 U32(mark(X)) -> mark(U32(X)) 1158.63/295.45 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1158.63/295.45 U42(mark(X1), X2) -> mark(U42(X1, X2)) 1158.63/295.45 U43(mark(X)) -> mark(U43(X)) 1158.63/295.45 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 1158.63/295.45 U52(mark(X1), X2) -> mark(U52(X1, X2)) 1158.63/295.45 U53(mark(X)) -> mark(U53(X)) 1158.63/295.45 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1158.63/295.45 s(mark(X)) -> mark(s(X)) 1158.63/295.45 length(mark(X)) -> mark(length(X)) 1158.63/295.45 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.63/295.45 proper(zeros) -> ok(zeros) 1158.63/295.45 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.63/295.45 proper(0) -> ok(0) 1158.63/295.45 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.63/295.45 proper(tt) -> ok(tt) 1158.63/295.45 proper(U12(X)) -> U12(proper(X)) 1158.63/295.45 proper(isNatList(X)) -> isNatList(proper(X)) 1158.63/295.45 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1158.63/295.45 proper(U22(X)) -> U22(proper(X)) 1158.63/295.45 proper(isNat(X)) -> isNat(proper(X)) 1158.63/295.45 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1158.63/295.45 proper(U32(X)) -> U32(proper(X)) 1158.63/295.45 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 1158.63/295.45 proper(U43(X)) -> U43(proper(X)) 1158.63/295.45 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.63/295.45 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 1158.63/295.45 proper(U53(X)) -> U53(proper(X)) 1158.63/295.45 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1158.63/295.45 proper(s(X)) -> s(proper(X)) 1158.63/295.45 proper(length(X)) -> length(proper(X)) 1158.63/295.45 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.63/295.45 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1158.63/295.45 proper(isNatKind(X)) -> isNatKind(proper(X)) 1158.63/295.45 proper(nil) -> ok(nil) 1158.63/295.45 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.63/295.45 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.63/295.45 U12(ok(X)) -> ok(U12(X)) 1158.63/295.45 isNatList(ok(X)) -> ok(isNatList(X)) 1158.63/295.45 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1158.63/295.45 U22(ok(X)) -> ok(U22(X)) 1158.63/295.45 isNat(ok(X)) -> ok(isNat(X)) 1158.63/295.45 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1158.63/295.45 U32(ok(X)) -> ok(U32(X)) 1158.63/295.45 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1158.63/295.45 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 1158.63/295.45 U43(ok(X)) -> ok(U43(X)) 1158.63/295.45 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.63/295.45 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 1158.63/295.45 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 1158.63/295.45 U53(ok(X)) -> ok(U53(X)) 1158.63/295.45 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1158.63/295.45 s(ok(X)) -> ok(s(X)) 1158.63/295.45 length(ok(X)) -> ok(length(X)) 1158.63/295.45 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.63/295.45 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1158.63/295.45 isNatKind(ok(X)) -> ok(isNatKind(X)) 1158.63/295.45 top(mark(X)) -> top(proper(X)) 1158.63/295.45 top(ok(X)) -> top(active(X)) 1158.63/295.45 1158.63/295.45 S is empty. 1158.63/295.45 Rewrite Strategy: FULL 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1158.63/295.45 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1158.63/295.45 1158.63/295.45 The rewrite sequence 1158.63/295.45 1158.63/295.45 U52(ok(X1), ok(X2)) ->^+ ok(U52(X1, X2)) 1158.63/295.45 1158.63/295.45 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1158.63/295.45 1158.63/295.45 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 1158.63/295.45 1158.63/295.45 The result substitution is [ ]. 1158.63/295.45 1158.63/295.45 1158.63/295.45 1158.63/295.45 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (4) 1158.63/295.45 Complex Obligation (BEST) 1158.63/295.45 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (5) 1158.63/295.45 Obligation: 1158.63/295.45 Proved the lower bound n^1 for the following obligation: 1158.63/295.45 1158.63/295.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.63/295.45 1158.63/295.45 1158.63/295.45 The TRS R consists of the following rules: 1158.63/295.45 1158.63/295.45 active(zeros) -> mark(cons(0, zeros)) 1158.63/295.45 active(U11(tt, V1)) -> mark(U12(isNatList(V1))) 1158.63/295.45 active(U12(tt)) -> mark(tt) 1158.63/295.45 active(U21(tt, V1)) -> mark(U22(isNat(V1))) 1158.63/295.45 active(U22(tt)) -> mark(tt) 1158.63/295.45 active(U31(tt, V)) -> mark(U32(isNatList(V))) 1158.63/295.45 active(U32(tt)) -> mark(tt) 1158.63/295.45 active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) 1158.63/295.45 active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) 1158.63/295.45 active(U43(tt)) -> mark(tt) 1158.63/295.45 active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) 1158.63/295.45 active(U52(tt, V2)) -> mark(U53(isNatList(V2))) 1158.63/295.45 active(U53(tt)) -> mark(tt) 1158.63/295.45 active(U61(tt, L)) -> mark(s(length(L))) 1158.63/295.45 active(and(tt, X)) -> mark(X) 1158.63/295.45 active(isNat(0)) -> mark(tt) 1158.63/295.45 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1158.63/295.45 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1158.63/295.45 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1158.63/295.45 active(isNatIList(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(isNatIListKind(nil)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) 1158.63/295.45 active(isNatKind(0)) -> mark(tt) 1158.63/295.45 active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) 1158.63/295.45 active(isNatKind(s(V1))) -> mark(isNatKind(V1)) 1158.63/295.45 active(isNatList(nil)) -> mark(tt) 1158.63/295.45 active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(length(nil)) -> mark(0) 1158.63/295.45 active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) 1158.63/295.45 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.63/295.45 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.63/295.45 active(U12(X)) -> U12(active(X)) 1158.63/295.45 active(U21(X1, X2)) -> U21(active(X1), X2) 1158.63/295.45 active(U22(X)) -> U22(active(X)) 1158.63/295.45 active(U31(X1, X2)) -> U31(active(X1), X2) 1158.63/295.45 active(U32(X)) -> U32(active(X)) 1158.63/295.45 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1158.63/295.45 active(U42(X1, X2)) -> U42(active(X1), X2) 1158.63/295.45 active(U43(X)) -> U43(active(X)) 1158.63/295.45 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 1158.63/295.45 active(U52(X1, X2)) -> U52(active(X1), X2) 1158.63/295.45 active(U53(X)) -> U53(active(X)) 1158.63/295.45 active(U61(X1, X2)) -> U61(active(X1), X2) 1158.63/295.45 active(s(X)) -> s(active(X)) 1158.63/295.45 active(length(X)) -> length(active(X)) 1158.63/295.45 active(and(X1, X2)) -> and(active(X1), X2) 1158.63/295.45 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.63/295.45 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.63/295.45 U12(mark(X)) -> mark(U12(X)) 1158.63/295.45 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1158.63/295.45 U22(mark(X)) -> mark(U22(X)) 1158.63/295.45 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1158.63/295.45 U32(mark(X)) -> mark(U32(X)) 1158.63/295.45 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1158.63/295.45 U42(mark(X1), X2) -> mark(U42(X1, X2)) 1158.63/295.45 U43(mark(X)) -> mark(U43(X)) 1158.63/295.45 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 1158.63/295.45 U52(mark(X1), X2) -> mark(U52(X1, X2)) 1158.63/295.45 U53(mark(X)) -> mark(U53(X)) 1158.63/295.45 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1158.63/295.45 s(mark(X)) -> mark(s(X)) 1158.63/295.45 length(mark(X)) -> mark(length(X)) 1158.63/295.45 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.63/295.45 proper(zeros) -> ok(zeros) 1158.63/295.45 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.63/295.45 proper(0) -> ok(0) 1158.63/295.45 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.63/295.45 proper(tt) -> ok(tt) 1158.63/295.45 proper(U12(X)) -> U12(proper(X)) 1158.63/295.45 proper(isNatList(X)) -> isNatList(proper(X)) 1158.63/295.45 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1158.63/295.45 proper(U22(X)) -> U22(proper(X)) 1158.63/295.45 proper(isNat(X)) -> isNat(proper(X)) 1158.63/295.45 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1158.63/295.45 proper(U32(X)) -> U32(proper(X)) 1158.63/295.45 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 1158.63/295.45 proper(U43(X)) -> U43(proper(X)) 1158.63/295.45 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.63/295.45 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 1158.63/295.45 proper(U53(X)) -> U53(proper(X)) 1158.63/295.45 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1158.63/295.45 proper(s(X)) -> s(proper(X)) 1158.63/295.45 proper(length(X)) -> length(proper(X)) 1158.63/295.45 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.63/295.45 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1158.63/295.45 proper(isNatKind(X)) -> isNatKind(proper(X)) 1158.63/295.45 proper(nil) -> ok(nil) 1158.63/295.45 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.63/295.45 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.63/295.45 U12(ok(X)) -> ok(U12(X)) 1158.63/295.45 isNatList(ok(X)) -> ok(isNatList(X)) 1158.63/295.45 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1158.63/295.45 U22(ok(X)) -> ok(U22(X)) 1158.63/295.45 isNat(ok(X)) -> ok(isNat(X)) 1158.63/295.45 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1158.63/295.45 U32(ok(X)) -> ok(U32(X)) 1158.63/295.45 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1158.63/295.45 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 1158.63/295.45 U43(ok(X)) -> ok(U43(X)) 1158.63/295.45 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.63/295.45 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 1158.63/295.45 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 1158.63/295.45 U53(ok(X)) -> ok(U53(X)) 1158.63/295.45 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1158.63/295.45 s(ok(X)) -> ok(s(X)) 1158.63/295.45 length(ok(X)) -> ok(length(X)) 1158.63/295.45 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.63/295.45 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1158.63/295.45 isNatKind(ok(X)) -> ok(isNatKind(X)) 1158.63/295.45 top(mark(X)) -> top(proper(X)) 1158.63/295.45 top(ok(X)) -> top(active(X)) 1158.63/295.45 1158.63/295.45 S is empty. 1158.63/295.45 Rewrite Strategy: FULL 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (6) LowerBoundPropagationProof (FINISHED) 1158.63/295.45 Propagated lower bound. 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (7) 1158.63/295.45 BOUNDS(n^1, INF) 1158.63/295.45 1158.63/295.45 ---------------------------------------- 1158.63/295.45 1158.63/295.45 (8) 1158.63/295.45 Obligation: 1158.63/295.45 Analyzing the following TRS for decreasing loops: 1158.63/295.45 1158.63/295.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.63/295.45 1158.63/295.45 1158.63/295.45 The TRS R consists of the following rules: 1158.63/295.45 1158.63/295.45 active(zeros) -> mark(cons(0, zeros)) 1158.63/295.45 active(U11(tt, V1)) -> mark(U12(isNatList(V1))) 1158.63/295.45 active(U12(tt)) -> mark(tt) 1158.63/295.45 active(U21(tt, V1)) -> mark(U22(isNat(V1))) 1158.63/295.45 active(U22(tt)) -> mark(tt) 1158.63/295.45 active(U31(tt, V)) -> mark(U32(isNatList(V))) 1158.63/295.45 active(U32(tt)) -> mark(tt) 1158.63/295.45 active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2)) 1158.63/295.45 active(U42(tt, V2)) -> mark(U43(isNatIList(V2))) 1158.63/295.45 active(U43(tt)) -> mark(tt) 1158.63/295.45 active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2)) 1158.63/295.45 active(U52(tt, V2)) -> mark(U53(isNatList(V2))) 1158.63/295.45 active(U53(tt)) -> mark(tt) 1158.63/295.45 active(U61(tt, L)) -> mark(s(length(L))) 1158.63/295.45 active(and(tt, X)) -> mark(X) 1158.63/295.45 active(isNat(0)) -> mark(tt) 1158.63/295.45 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1158.63/295.45 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1158.63/295.45 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1158.63/295.45 active(isNatIList(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(isNatIListKind(nil)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(zeros)) -> mark(tt) 1158.63/295.45 active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) 1158.63/295.45 active(isNatKind(0)) -> mark(tt) 1158.63/295.45 active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) 1158.63/295.45 active(isNatKind(s(V1))) -> mark(isNatKind(V1)) 1158.63/295.45 active(isNatList(nil)) -> mark(tt) 1158.63/295.45 active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) 1158.63/295.45 active(length(nil)) -> mark(0) 1158.63/295.45 active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) 1158.63/295.45 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.63/295.45 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.63/295.45 active(U12(X)) -> U12(active(X)) 1158.63/295.45 active(U21(X1, X2)) -> U21(active(X1), X2) 1158.63/295.45 active(U22(X)) -> U22(active(X)) 1158.63/295.45 active(U31(X1, X2)) -> U31(active(X1), X2) 1158.63/295.45 active(U32(X)) -> U32(active(X)) 1158.63/295.45 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1158.63/295.45 active(U42(X1, X2)) -> U42(active(X1), X2) 1158.63/295.45 active(U43(X)) -> U43(active(X)) 1158.63/295.45 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 1158.63/295.45 active(U52(X1, X2)) -> U52(active(X1), X2) 1158.63/295.45 active(U53(X)) -> U53(active(X)) 1158.63/295.45 active(U61(X1, X2)) -> U61(active(X1), X2) 1158.63/295.45 active(s(X)) -> s(active(X)) 1158.63/295.45 active(length(X)) -> length(active(X)) 1158.63/295.45 active(and(X1, X2)) -> and(active(X1), X2) 1158.63/295.45 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.63/295.45 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.63/295.45 U12(mark(X)) -> mark(U12(X)) 1158.63/295.45 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1158.63/295.45 U22(mark(X)) -> mark(U22(X)) 1158.63/295.45 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1158.63/295.45 U32(mark(X)) -> mark(U32(X)) 1158.63/295.45 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1158.63/295.45 U42(mark(X1), X2) -> mark(U42(X1, X2)) 1158.63/295.45 U43(mark(X)) -> mark(U43(X)) 1158.63/295.45 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 1158.63/295.45 U52(mark(X1), X2) -> mark(U52(X1, X2)) 1158.63/295.45 U53(mark(X)) -> mark(U53(X)) 1158.63/295.45 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1158.63/295.45 s(mark(X)) -> mark(s(X)) 1158.63/295.45 length(mark(X)) -> mark(length(X)) 1158.63/295.45 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.63/295.45 proper(zeros) -> ok(zeros) 1158.63/295.45 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.63/295.45 proper(0) -> ok(0) 1158.63/295.45 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.63/295.45 proper(tt) -> ok(tt) 1158.63/295.45 proper(U12(X)) -> U12(proper(X)) 1158.63/295.45 proper(isNatList(X)) -> isNatList(proper(X)) 1158.63/295.45 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1158.63/295.45 proper(U22(X)) -> U22(proper(X)) 1158.63/295.45 proper(isNat(X)) -> isNat(proper(X)) 1158.63/295.45 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1158.63/295.45 proper(U32(X)) -> U32(proper(X)) 1158.63/295.45 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 1158.63/295.45 proper(U43(X)) -> U43(proper(X)) 1158.63/295.45 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.63/295.45 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 1158.63/295.45 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 1158.63/295.45 proper(U53(X)) -> U53(proper(X)) 1158.63/295.45 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1158.63/295.45 proper(s(X)) -> s(proper(X)) 1158.63/295.45 proper(length(X)) -> length(proper(X)) 1158.63/295.45 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.63/295.45 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1158.63/295.45 proper(isNatKind(X)) -> isNatKind(proper(X)) 1158.63/295.45 proper(nil) -> ok(nil) 1158.63/295.45 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.63/295.45 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.63/295.45 U12(ok(X)) -> ok(U12(X)) 1158.63/295.45 isNatList(ok(X)) -> ok(isNatList(X)) 1158.63/295.45 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1158.63/295.45 U22(ok(X)) -> ok(U22(X)) 1158.63/295.45 isNat(ok(X)) -> ok(isNat(X)) 1158.63/295.45 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1158.63/295.45 U32(ok(X)) -> ok(U32(X)) 1158.63/295.45 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1158.63/295.45 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 1158.63/295.45 U43(ok(X)) -> ok(U43(X)) 1158.63/295.45 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.63/295.45 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 1158.63/295.45 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 1158.63/295.45 U53(ok(X)) -> ok(U53(X)) 1158.63/295.45 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1158.63/295.45 s(ok(X)) -> ok(s(X)) 1158.63/295.45 length(ok(X)) -> ok(length(X)) 1158.63/295.45 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.63/295.45 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1158.63/295.45 isNatKind(ok(X)) -> ok(isNatKind(X)) 1158.63/295.45 top(mark(X)) -> top(proper(X)) 1158.63/295.45 top(ok(X)) -> top(active(X)) 1158.63/295.45 1158.63/295.45 S is empty. 1158.63/295.45 Rewrite Strategy: FULL 1158.97/295.53 EOF