4.77/2.05 WORST_CASE(NON_POLY, ?) 5.16/2.05 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.16/2.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.16/2.05 5.16/2.05 5.16/2.05 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.16/2.05 5.16/2.05 (0) CpxTRS 5.16/2.05 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 5.16/2.05 (2) TRS for Loop Detection 5.16/2.05 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 5.16/2.05 (4) BEST 5.16/2.05 (5) proven lower bound 5.16/2.05 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 5.16/2.05 (7) BOUNDS(n^1, INF) 5.16/2.05 (8) TRS for Loop Detection 5.16/2.05 (9) DecreasingLoopProof [FINISHED, 343 ms] 5.16/2.05 (10) BOUNDS(EXP, INF) 5.16/2.05 5.16/2.05 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (0) 5.16/2.05 Obligation: 5.16/2.05 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.16/2.05 5.16/2.05 5.16/2.05 The TRS R consists of the following rules: 5.16/2.05 5.16/2.05 zeros -> cons(0, n__zeros) 5.16/2.05 U11(tt, L) -> s(length(activate(L))) 5.16/2.05 and(tt, X) -> activate(X) 5.16/2.05 isNat(n__0) -> tt 5.16/2.05 isNat(n__length(V1)) -> isNatList(activate(V1)) 5.16/2.05 isNat(n__s(V1)) -> isNat(activate(V1)) 5.16/2.05 isNatIList(V) -> isNatList(activate(V)) 5.16/2.05 isNatIList(n__zeros) -> tt 5.16/2.05 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 5.16/2.05 isNatList(n__nil) -> tt 5.16/2.05 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 5.16/2.05 length(nil) -> 0 5.16/2.05 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 5.16/2.05 zeros -> n__zeros 5.16/2.05 0 -> n__0 5.16/2.05 length(X) -> n__length(X) 5.16/2.05 s(X) -> n__s(X) 5.16/2.05 cons(X1, X2) -> n__cons(X1, X2) 5.16/2.05 isNatIList(X) -> n__isNatIList(X) 5.16/2.05 nil -> n__nil 5.16/2.05 isNatList(X) -> n__isNatList(X) 5.16/2.05 isNat(X) -> n__isNat(X) 5.16/2.05 activate(n__zeros) -> zeros 5.16/2.05 activate(n__0) -> 0 5.16/2.05 activate(n__length(X)) -> length(activate(X)) 5.16/2.05 activate(n__s(X)) -> s(activate(X)) 5.16/2.05 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.16/2.05 activate(n__isNatIList(X)) -> isNatIList(X) 5.16/2.05 activate(n__nil) -> nil 5.16/2.05 activate(n__isNatList(X)) -> isNatList(X) 5.16/2.05 activate(n__isNat(X)) -> isNat(X) 5.16/2.05 activate(X) -> X 5.16/2.05 5.16/2.05 S is empty. 5.16/2.05 Rewrite Strategy: FULL 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 5.16/2.05 Transformed a relative TRS into a decreasing-loop problem. 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (2) 5.16/2.05 Obligation: 5.16/2.05 Analyzing the following TRS for decreasing loops: 5.16/2.05 5.16/2.05 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.16/2.05 5.16/2.05 5.16/2.05 The TRS R consists of the following rules: 5.16/2.05 5.16/2.05 zeros -> cons(0, n__zeros) 5.16/2.05 U11(tt, L) -> s(length(activate(L))) 5.16/2.05 and(tt, X) -> activate(X) 5.16/2.05 isNat(n__0) -> tt 5.16/2.05 isNat(n__length(V1)) -> isNatList(activate(V1)) 5.16/2.05 isNat(n__s(V1)) -> isNat(activate(V1)) 5.16/2.05 isNatIList(V) -> isNatList(activate(V)) 5.16/2.05 isNatIList(n__zeros) -> tt 5.16/2.05 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 5.16/2.05 isNatList(n__nil) -> tt 5.16/2.05 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 5.16/2.05 length(nil) -> 0 5.16/2.05 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 5.16/2.05 zeros -> n__zeros 5.16/2.05 0 -> n__0 5.16/2.05 length(X) -> n__length(X) 5.16/2.05 s(X) -> n__s(X) 5.16/2.05 cons(X1, X2) -> n__cons(X1, X2) 5.16/2.05 isNatIList(X) -> n__isNatIList(X) 5.16/2.05 nil -> n__nil 5.16/2.05 isNatList(X) -> n__isNatList(X) 5.16/2.05 isNat(X) -> n__isNat(X) 5.16/2.05 activate(n__zeros) -> zeros 5.16/2.05 activate(n__0) -> 0 5.16/2.05 activate(n__length(X)) -> length(activate(X)) 5.16/2.05 activate(n__s(X)) -> s(activate(X)) 5.16/2.05 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.16/2.05 activate(n__isNatIList(X)) -> isNatIList(X) 5.16/2.05 activate(n__nil) -> nil 5.16/2.05 activate(n__isNatList(X)) -> isNatList(X) 5.16/2.05 activate(n__isNat(X)) -> isNat(X) 5.16/2.05 activate(X) -> X 5.16/2.05 5.16/2.05 S is empty. 5.16/2.05 Rewrite Strategy: FULL 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (3) DecreasingLoopProof (LOWER BOUND(ID)) 5.16/2.05 The following loop(s) give(s) rise to the lower bound Omega(n^1): 5.16/2.05 5.16/2.05 The rewrite sequence 5.16/2.05 5.16/2.05 activate(n__s(X)) ->^+ s(activate(X)) 5.16/2.05 5.16/2.05 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 5.16/2.05 5.16/2.05 The pumping substitution is [X / n__s(X)]. 5.16/2.05 5.16/2.05 The result substitution is [ ]. 5.16/2.05 5.16/2.05 5.16/2.05 5.16/2.05 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (4) 5.16/2.05 Complex Obligation (BEST) 5.16/2.05 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (5) 5.16/2.05 Obligation: 5.16/2.05 Proved the lower bound n^1 for the following obligation: 5.16/2.05 5.16/2.05 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.16/2.05 5.16/2.05 5.16/2.05 The TRS R consists of the following rules: 5.16/2.05 5.16/2.05 zeros -> cons(0, n__zeros) 5.16/2.05 U11(tt, L) -> s(length(activate(L))) 5.16/2.05 and(tt, X) -> activate(X) 5.16/2.05 isNat(n__0) -> tt 5.16/2.05 isNat(n__length(V1)) -> isNatList(activate(V1)) 5.16/2.05 isNat(n__s(V1)) -> isNat(activate(V1)) 5.16/2.05 isNatIList(V) -> isNatList(activate(V)) 5.16/2.05 isNatIList(n__zeros) -> tt 5.16/2.05 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 5.16/2.05 isNatList(n__nil) -> tt 5.16/2.05 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 5.16/2.05 length(nil) -> 0 5.16/2.05 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 5.16/2.05 zeros -> n__zeros 5.16/2.05 0 -> n__0 5.16/2.05 length(X) -> n__length(X) 5.16/2.05 s(X) -> n__s(X) 5.16/2.05 cons(X1, X2) -> n__cons(X1, X2) 5.16/2.05 isNatIList(X) -> n__isNatIList(X) 5.16/2.05 nil -> n__nil 5.16/2.05 isNatList(X) -> n__isNatList(X) 5.16/2.05 isNat(X) -> n__isNat(X) 5.16/2.05 activate(n__zeros) -> zeros 5.16/2.05 activate(n__0) -> 0 5.16/2.05 activate(n__length(X)) -> length(activate(X)) 5.16/2.05 activate(n__s(X)) -> s(activate(X)) 5.16/2.05 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.16/2.05 activate(n__isNatIList(X)) -> isNatIList(X) 5.16/2.05 activate(n__nil) -> nil 5.16/2.05 activate(n__isNatList(X)) -> isNatList(X) 5.16/2.05 activate(n__isNat(X)) -> isNat(X) 5.16/2.05 activate(X) -> X 5.16/2.05 5.16/2.05 S is empty. 5.16/2.05 Rewrite Strategy: FULL 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (6) LowerBoundPropagationProof (FINISHED) 5.16/2.05 Propagated lower bound. 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (7) 5.16/2.05 BOUNDS(n^1, INF) 5.16/2.05 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (8) 5.16/2.05 Obligation: 5.16/2.05 Analyzing the following TRS for decreasing loops: 5.16/2.05 5.16/2.05 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.16/2.05 5.16/2.05 5.16/2.05 The TRS R consists of the following rules: 5.16/2.05 5.16/2.05 zeros -> cons(0, n__zeros) 5.16/2.05 U11(tt, L) -> s(length(activate(L))) 5.16/2.05 and(tt, X) -> activate(X) 5.16/2.05 isNat(n__0) -> tt 5.16/2.05 isNat(n__length(V1)) -> isNatList(activate(V1)) 5.16/2.05 isNat(n__s(V1)) -> isNat(activate(V1)) 5.16/2.05 isNatIList(V) -> isNatList(activate(V)) 5.16/2.05 isNatIList(n__zeros) -> tt 5.16/2.05 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 5.16/2.05 isNatList(n__nil) -> tt 5.16/2.05 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 5.16/2.05 length(nil) -> 0 5.16/2.05 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 5.16/2.05 zeros -> n__zeros 5.16/2.05 0 -> n__0 5.16/2.05 length(X) -> n__length(X) 5.16/2.05 s(X) -> n__s(X) 5.16/2.05 cons(X1, X2) -> n__cons(X1, X2) 5.16/2.05 isNatIList(X) -> n__isNatIList(X) 5.16/2.05 nil -> n__nil 5.16/2.05 isNatList(X) -> n__isNatList(X) 5.16/2.05 isNat(X) -> n__isNat(X) 5.16/2.05 activate(n__zeros) -> zeros 5.16/2.05 activate(n__0) -> 0 5.16/2.05 activate(n__length(X)) -> length(activate(X)) 5.16/2.05 activate(n__s(X)) -> s(activate(X)) 5.16/2.05 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.16/2.05 activate(n__isNatIList(X)) -> isNatIList(X) 5.16/2.05 activate(n__nil) -> nil 5.16/2.05 activate(n__isNatList(X)) -> isNatList(X) 5.16/2.05 activate(n__isNat(X)) -> isNat(X) 5.16/2.05 activate(X) -> X 5.16/2.05 5.16/2.05 S is empty. 5.16/2.05 Rewrite Strategy: FULL 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (9) DecreasingLoopProof (FINISHED) 5.16/2.05 The following loop(s) give(s) rise to the lower bound EXP: 5.16/2.05 5.16/2.05 The rewrite sequence 5.16/2.05 5.16/2.05 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U11(and(isNatList(activate(X22_0)), n__isNat(activate(X11_0))), activate(X22_0)) 5.16/2.05 5.16/2.05 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0]. 5.16/2.05 5.16/2.05 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 5.16/2.05 5.16/2.05 The result substitution is [ ]. 5.16/2.05 5.16/2.05 5.16/2.05 5.16/2.05 The rewrite sequence 5.16/2.05 5.16/2.05 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U11(and(isNatList(activate(X22_0)), n__isNat(activate(X11_0))), activate(X22_0)) 5.16/2.05 5.16/2.05 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 5.16/2.05 5.16/2.05 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 5.16/2.05 5.16/2.05 The result substitution is [ ]. 5.16/2.05 5.16/2.05 5.16/2.05 5.16/2.05 5.16/2.05 ---------------------------------------- 5.16/2.05 5.16/2.05 (10) 5.16/2.05 BOUNDS(EXP, INF) 5.16/2.09 EOF