456.95/291.64 WORST_CASE(Omega(n^1), ?) 456.95/291.64 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 456.95/291.64 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 456.95/291.64 456.95/291.64 456.95/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 456.95/291.64 456.95/291.64 (0) CpxTRS 456.95/291.64 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 456.95/291.64 (2) TRS for Loop Detection 456.95/291.64 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 456.95/291.64 (4) BEST 456.95/291.64 (5) proven lower bound 456.95/291.64 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 456.95/291.64 (7) BOUNDS(n^1, INF) 456.95/291.64 (8) TRS for Loop Detection 456.95/291.64 456.95/291.64 456.95/291.64 ---------------------------------------- 456.95/291.64 456.95/291.64 (0) 456.95/291.64 Obligation: 456.95/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 456.95/291.64 456.95/291.64 456.95/291.64 The TRS R consists of the following rules: 456.95/291.64 456.95/291.64 a__zeros -> cons(0, zeros) 456.95/291.64 a__U11(tt) -> tt 456.95/291.64 a__U21(tt) -> tt 456.95/291.64 a__U31(tt) -> tt 456.95/291.64 a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) 456.95/291.64 a__U42(tt) -> tt 456.95/291.64 a__U51(tt, V2) -> a__U52(a__isNatList(V2)) 456.95/291.64 a__U52(tt) -> tt 456.95/291.64 a__U61(tt, L, N) -> a__U62(a__isNat(N), L) 456.95/291.64 a__U62(tt, L) -> s(a__length(mark(L))) 456.95/291.64 a__isNat(0) -> tt 456.95/291.64 a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) 456.95/291.64 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 456.95/291.64 a__isNatIList(V) -> a__U31(a__isNatList(V)) 456.95/291.64 a__isNatIList(zeros) -> tt 456.95/291.64 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) 456.95/291.64 a__isNatList(nil) -> tt 456.95/291.64 a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) 456.95/291.64 a__length(nil) -> 0 456.95/291.64 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) 456.95/291.64 mark(zeros) -> a__zeros 456.95/291.64 mark(U11(X)) -> a__U11(mark(X)) 456.95/291.64 mark(U21(X)) -> a__U21(mark(X)) 456.95/291.64 mark(U31(X)) -> a__U31(mark(X)) 456.95/291.64 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 456.95/291.64 mark(U42(X)) -> a__U42(mark(X)) 456.95/291.64 mark(isNatIList(X)) -> a__isNatIList(X) 456.95/291.64 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 456.95/291.64 mark(U52(X)) -> a__U52(mark(X)) 456.95/291.64 mark(isNatList(X)) -> a__isNatList(X) 456.95/291.64 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 456.95/291.64 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 456.95/291.64 mark(isNat(X)) -> a__isNat(X) 456.95/291.64 mark(length(X)) -> a__length(mark(X)) 456.95/291.64 mark(cons(X1, X2)) -> cons(mark(X1), X2) 456.95/291.64 mark(0) -> 0 456.95/291.64 mark(tt) -> tt 456.95/291.64 mark(s(X)) -> s(mark(X)) 456.95/291.64 mark(nil) -> nil 456.95/291.64 a__zeros -> zeros 456.95/291.64 a__U11(X) -> U11(X) 456.95/291.64 a__U21(X) -> U21(X) 456.95/291.64 a__U31(X) -> U31(X) 456.95/291.64 a__U41(X1, X2) -> U41(X1, X2) 456.95/291.64 a__U42(X) -> U42(X) 456.95/291.64 a__isNatIList(X) -> isNatIList(X) 456.95/291.64 a__U51(X1, X2) -> U51(X1, X2) 456.95/291.64 a__U52(X) -> U52(X) 456.95/291.64 a__isNatList(X) -> isNatList(X) 456.95/291.64 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 456.95/291.64 a__U62(X1, X2) -> U62(X1, X2) 456.95/291.64 a__isNat(X) -> isNat(X) 456.95/291.64 a__length(X) -> length(X) 456.95/291.64 456.95/291.64 S is empty. 456.95/291.64 Rewrite Strategy: FULL 456.95/291.64 ---------------------------------------- 456.95/291.64 456.95/291.64 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 456.95/291.64 Transformed a relative TRS into a decreasing-loop problem. 456.95/291.64 ---------------------------------------- 456.95/291.64 456.95/291.64 (2) 456.95/291.64 Obligation: 456.95/291.64 Analyzing the following TRS for decreasing loops: 456.95/291.64 456.95/291.64 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 456.95/291.64 456.95/291.64 456.95/291.64 The TRS R consists of the following rules: 456.95/291.64 456.95/291.64 a__zeros -> cons(0, zeros) 456.95/291.64 a__U11(tt) -> tt 456.95/291.64 a__U21(tt) -> tt 456.95/291.64 a__U31(tt) -> tt 456.95/291.64 a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) 456.95/291.64 a__U42(tt) -> tt 456.95/291.64 a__U51(tt, V2) -> a__U52(a__isNatList(V2)) 456.95/291.64 a__U52(tt) -> tt 456.95/291.64 a__U61(tt, L, N) -> a__U62(a__isNat(N), L) 456.95/291.64 a__U62(tt, L) -> s(a__length(mark(L))) 456.95/291.64 a__isNat(0) -> tt 456.95/291.64 a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) 456.95/291.64 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 456.95/291.64 a__isNatIList(V) -> a__U31(a__isNatList(V)) 456.95/291.64 a__isNatIList(zeros) -> tt 456.95/291.64 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) 456.95/291.64 a__isNatList(nil) -> tt 456.95/291.64 a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) 456.95/291.64 a__length(nil) -> 0 456.95/291.64 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) 456.95/291.64 mark(zeros) -> a__zeros 456.95/291.64 mark(U11(X)) -> a__U11(mark(X)) 456.95/291.64 mark(U21(X)) -> a__U21(mark(X)) 456.95/291.64 mark(U31(X)) -> a__U31(mark(X)) 456.95/291.64 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 456.95/291.64 mark(U42(X)) -> a__U42(mark(X)) 456.95/291.64 mark(isNatIList(X)) -> a__isNatIList(X) 456.95/291.64 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 456.95/291.64 mark(U52(X)) -> a__U52(mark(X)) 456.95/291.64 mark(isNatList(X)) -> a__isNatList(X) 456.95/291.64 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 456.95/291.64 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 456.95/291.64 mark(isNat(X)) -> a__isNat(X) 456.95/291.64 mark(length(X)) -> a__length(mark(X)) 456.95/291.64 mark(cons(X1, X2)) -> cons(mark(X1), X2) 456.95/291.64 mark(0) -> 0 456.95/291.64 mark(tt) -> tt 456.95/291.64 mark(s(X)) -> s(mark(X)) 456.95/291.64 mark(nil) -> nil 456.95/291.64 a__zeros -> zeros 456.95/291.64 a__U11(X) -> U11(X) 456.95/291.64 a__U21(X) -> U21(X) 456.95/291.64 a__U31(X) -> U31(X) 456.95/291.64 a__U41(X1, X2) -> U41(X1, X2) 456.95/291.64 a__U42(X) -> U42(X) 456.95/291.64 a__isNatIList(X) -> isNatIList(X) 456.95/291.64 a__U51(X1, X2) -> U51(X1, X2) 456.95/291.64 a__U52(X) -> U52(X) 456.95/291.64 a__isNatList(X) -> isNatList(X) 456.95/291.64 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 456.95/291.64 a__U62(X1, X2) -> U62(X1, X2) 456.95/291.64 a__isNat(X) -> isNat(X) 456.95/291.64 a__length(X) -> length(X) 456.95/291.64 456.95/291.64 S is empty. 456.95/291.64 Rewrite Strategy: FULL 456.95/291.64 ---------------------------------------- 456.95/291.64 456.95/291.64 (3) DecreasingLoopProof (LOWER BOUND(ID)) 456.95/291.64 The following loop(s) give(s) rise to the lower bound Omega(n^1): 456.95/291.64 456.95/291.64 The rewrite sequence 456.95/291.64 456.95/291.64 mark(U61(X1, X2, X3)) ->^+ a__U61(mark(X1), X2, X3) 456.95/291.64 456.95/291.64 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 456.95/291.65 456.95/291.65 The pumping substitution is [X1 / U61(X1, X2, X3)]. 456.95/291.65 456.95/291.65 The result substitution is [ ]. 456.95/291.65 456.95/291.65 456.95/291.65 456.95/291.65 456.95/291.65 ---------------------------------------- 456.95/291.65 456.95/291.65 (4) 456.95/291.65 Complex Obligation (BEST) 456.95/291.65 456.95/291.65 ---------------------------------------- 456.95/291.65 456.95/291.65 (5) 456.95/291.65 Obligation: 456.95/291.65 Proved the lower bound n^1 for the following obligation: 456.95/291.65 456.95/291.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 456.95/291.65 456.95/291.65 456.95/291.65 The TRS R consists of the following rules: 456.95/291.65 456.95/291.65 a__zeros -> cons(0, zeros) 456.95/291.65 a__U11(tt) -> tt 456.95/291.65 a__U21(tt) -> tt 456.95/291.65 a__U31(tt) -> tt 456.95/291.65 a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) 456.95/291.65 a__U42(tt) -> tt 456.95/291.65 a__U51(tt, V2) -> a__U52(a__isNatList(V2)) 456.95/291.65 a__U52(tt) -> tt 456.95/291.65 a__U61(tt, L, N) -> a__U62(a__isNat(N), L) 456.95/291.65 a__U62(tt, L) -> s(a__length(mark(L))) 456.95/291.65 a__isNat(0) -> tt 456.95/291.65 a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) 456.95/291.65 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 456.95/291.65 a__isNatIList(V) -> a__U31(a__isNatList(V)) 456.95/291.65 a__isNatIList(zeros) -> tt 456.95/291.65 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) 456.95/291.65 a__isNatList(nil) -> tt 456.95/291.65 a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) 456.95/291.65 a__length(nil) -> 0 456.95/291.65 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) 456.95/291.65 mark(zeros) -> a__zeros 456.95/291.65 mark(U11(X)) -> a__U11(mark(X)) 456.95/291.65 mark(U21(X)) -> a__U21(mark(X)) 456.95/291.65 mark(U31(X)) -> a__U31(mark(X)) 456.95/291.65 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 456.95/291.65 mark(U42(X)) -> a__U42(mark(X)) 456.95/291.65 mark(isNatIList(X)) -> a__isNatIList(X) 456.95/291.65 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 456.95/291.65 mark(U52(X)) -> a__U52(mark(X)) 456.95/291.65 mark(isNatList(X)) -> a__isNatList(X) 456.95/291.65 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 456.95/291.65 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 456.95/291.65 mark(isNat(X)) -> a__isNat(X) 456.95/291.65 mark(length(X)) -> a__length(mark(X)) 456.95/291.65 mark(cons(X1, X2)) -> cons(mark(X1), X2) 456.95/291.65 mark(0) -> 0 456.95/291.65 mark(tt) -> tt 456.95/291.65 mark(s(X)) -> s(mark(X)) 456.95/291.65 mark(nil) -> nil 456.95/291.65 a__zeros -> zeros 456.95/291.65 a__U11(X) -> U11(X) 456.95/291.65 a__U21(X) -> U21(X) 456.95/291.65 a__U31(X) -> U31(X) 456.95/291.65 a__U41(X1, X2) -> U41(X1, X2) 456.95/291.65 a__U42(X) -> U42(X) 456.95/291.65 a__isNatIList(X) -> isNatIList(X) 456.95/291.65 a__U51(X1, X2) -> U51(X1, X2) 456.95/291.65 a__U52(X) -> U52(X) 456.95/291.65 a__isNatList(X) -> isNatList(X) 456.95/291.65 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 456.95/291.65 a__U62(X1, X2) -> U62(X1, X2) 456.95/291.65 a__isNat(X) -> isNat(X) 456.95/291.65 a__length(X) -> length(X) 456.95/291.65 456.95/291.65 S is empty. 456.95/291.65 Rewrite Strategy: FULL 456.95/291.65 ---------------------------------------- 456.95/291.65 456.95/291.65 (6) LowerBoundPropagationProof (FINISHED) 456.95/291.65 Propagated lower bound. 456.95/291.65 ---------------------------------------- 456.95/291.65 456.95/291.65 (7) 456.95/291.65 BOUNDS(n^1, INF) 456.95/291.65 456.95/291.65 ---------------------------------------- 456.95/291.65 456.95/291.65 (8) 456.95/291.65 Obligation: 456.95/291.65 Analyzing the following TRS for decreasing loops: 456.95/291.65 456.95/291.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 456.95/291.65 456.95/291.65 456.95/291.65 The TRS R consists of the following rules: 456.95/291.65 456.95/291.65 a__zeros -> cons(0, zeros) 456.95/291.65 a__U11(tt) -> tt 456.95/291.65 a__U21(tt) -> tt 456.95/291.65 a__U31(tt) -> tt 456.95/291.65 a__U41(tt, V2) -> a__U42(a__isNatIList(V2)) 456.95/291.65 a__U42(tt) -> tt 456.95/291.65 a__U51(tt, V2) -> a__U52(a__isNatList(V2)) 456.95/291.65 a__U52(tt) -> tt 456.95/291.65 a__U61(tt, L, N) -> a__U62(a__isNat(N), L) 456.95/291.65 a__U62(tt, L) -> s(a__length(mark(L))) 456.95/291.65 a__isNat(0) -> tt 456.95/291.65 a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) 456.95/291.65 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 456.95/291.65 a__isNatIList(V) -> a__U31(a__isNatList(V)) 456.95/291.65 a__isNatIList(zeros) -> tt 456.95/291.65 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2) 456.95/291.65 a__isNatList(nil) -> tt 456.95/291.65 a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2) 456.95/291.65 a__length(nil) -> 0 456.95/291.65 a__length(cons(N, L)) -> a__U61(a__isNatList(L), L, N) 456.95/291.65 mark(zeros) -> a__zeros 456.95/291.65 mark(U11(X)) -> a__U11(mark(X)) 456.95/291.65 mark(U21(X)) -> a__U21(mark(X)) 456.95/291.65 mark(U31(X)) -> a__U31(mark(X)) 456.95/291.65 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 456.95/291.65 mark(U42(X)) -> a__U42(mark(X)) 456.95/291.65 mark(isNatIList(X)) -> a__isNatIList(X) 456.95/291.65 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 456.95/291.65 mark(U52(X)) -> a__U52(mark(X)) 456.95/291.65 mark(isNatList(X)) -> a__isNatList(X) 456.95/291.65 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 456.95/291.65 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 456.95/291.65 mark(isNat(X)) -> a__isNat(X) 456.95/291.65 mark(length(X)) -> a__length(mark(X)) 456.95/291.65 mark(cons(X1, X2)) -> cons(mark(X1), X2) 456.95/291.65 mark(0) -> 0 456.95/291.65 mark(tt) -> tt 456.95/291.65 mark(s(X)) -> s(mark(X)) 456.95/291.65 mark(nil) -> nil 456.95/291.65 a__zeros -> zeros 456.95/291.65 a__U11(X) -> U11(X) 456.95/291.65 a__U21(X) -> U21(X) 456.95/291.65 a__U31(X) -> U31(X) 456.95/291.65 a__U41(X1, X2) -> U41(X1, X2) 456.95/291.65 a__U42(X) -> U42(X) 456.95/291.65 a__isNatIList(X) -> isNatIList(X) 456.95/291.65 a__U51(X1, X2) -> U51(X1, X2) 456.95/291.65 a__U52(X) -> U52(X) 456.95/291.65 a__isNatList(X) -> isNatList(X) 456.95/291.65 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 456.95/291.65 a__U62(X1, X2) -> U62(X1, X2) 456.95/291.65 a__isNat(X) -> isNat(X) 456.95/291.65 a__length(X) -> length(X) 456.95/291.65 456.95/291.65 S is empty. 456.95/291.65 Rewrite Strategy: FULL 457.22/291.68 EOF