5.14/2.19 WORST_CASE(NON_POLY, ?) 5.57/2.20 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.57/2.20 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.57/2.20 5.57/2.20 5.57/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.57/2.20 5.57/2.20 (0) CpxTRS 5.57/2.20 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 5.57/2.20 (2) TRS for Loop Detection 5.57/2.20 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 5.57/2.20 (4) BEST 5.57/2.20 (5) proven lower bound 5.57/2.20 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 5.57/2.20 (7) BOUNDS(n^1, INF) 5.57/2.20 (8) TRS for Loop Detection 5.57/2.20 (9) DecreasingLoopProof [FINISHED, 391 ms] 5.57/2.20 (10) BOUNDS(EXP, INF) 5.57/2.20 5.57/2.20 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (0) 5.57/2.20 Obligation: 5.57/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.57/2.20 5.57/2.20 5.57/2.20 The TRS R consists of the following rules: 5.57/2.20 5.57/2.20 and(tt, T) -> T 5.57/2.20 isNatIList(IL) -> isNatList(activate(IL)) 5.57/2.20 isNat(n__0) -> tt 5.57/2.20 isNat(n__s(N)) -> isNat(activate(N)) 5.57/2.20 isNat(n__length(L)) -> isNatList(activate(L)) 5.57/2.20 isNatIList(n__zeros) -> tt 5.57/2.20 isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 isNatList(n__nil) -> tt 5.57/2.20 isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) 5.57/2.20 isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 zeros -> cons(0, n__zeros) 5.57/2.20 take(0, IL) -> uTake1(isNatIList(IL)) 5.57/2.20 uTake1(tt) -> nil 5.57/2.20 take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) 5.57/2.20 uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) 5.57/2.20 length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) 5.57/2.20 uLength(tt, L) -> s(length(activate(L))) 5.57/2.20 0 -> n__0 5.57/2.20 s(X) -> n__s(X) 5.57/2.20 length(X) -> n__length(X) 5.57/2.20 zeros -> n__zeros 5.57/2.20 cons(X1, X2) -> n__cons(X1, X2) 5.57/2.20 nil -> n__nil 5.57/2.20 take(X1, X2) -> n__take(X1, X2) 5.57/2.20 activate(n__0) -> 0 5.57/2.20 activate(n__s(X)) -> s(activate(X)) 5.57/2.20 activate(n__length(X)) -> length(activate(X)) 5.57/2.20 activate(n__zeros) -> zeros 5.57/2.20 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.57/2.20 activate(n__nil) -> nil 5.57/2.20 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 5.57/2.20 activate(X) -> X 5.57/2.20 5.57/2.20 S is empty. 5.57/2.20 Rewrite Strategy: FULL 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 5.57/2.20 Transformed a relative TRS into a decreasing-loop problem. 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (2) 5.57/2.20 Obligation: 5.57/2.20 Analyzing the following TRS for decreasing loops: 5.57/2.20 5.57/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.57/2.20 5.57/2.20 5.57/2.20 The TRS R consists of the following rules: 5.57/2.20 5.57/2.20 and(tt, T) -> T 5.57/2.20 isNatIList(IL) -> isNatList(activate(IL)) 5.57/2.20 isNat(n__0) -> tt 5.57/2.20 isNat(n__s(N)) -> isNat(activate(N)) 5.57/2.20 isNat(n__length(L)) -> isNatList(activate(L)) 5.57/2.20 isNatIList(n__zeros) -> tt 5.57/2.20 isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 isNatList(n__nil) -> tt 5.57/2.20 isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) 5.57/2.20 isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 zeros -> cons(0, n__zeros) 5.57/2.20 take(0, IL) -> uTake1(isNatIList(IL)) 5.57/2.20 uTake1(tt) -> nil 5.57/2.20 take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) 5.57/2.20 uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) 5.57/2.20 length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) 5.57/2.20 uLength(tt, L) -> s(length(activate(L))) 5.57/2.20 0 -> n__0 5.57/2.20 s(X) -> n__s(X) 5.57/2.20 length(X) -> n__length(X) 5.57/2.20 zeros -> n__zeros 5.57/2.20 cons(X1, X2) -> n__cons(X1, X2) 5.57/2.20 nil -> n__nil 5.57/2.20 take(X1, X2) -> n__take(X1, X2) 5.57/2.20 activate(n__0) -> 0 5.57/2.20 activate(n__s(X)) -> s(activate(X)) 5.57/2.20 activate(n__length(X)) -> length(activate(X)) 5.57/2.20 activate(n__zeros) -> zeros 5.57/2.20 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.57/2.20 activate(n__nil) -> nil 5.57/2.20 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 5.57/2.20 activate(X) -> X 5.57/2.20 5.57/2.20 S is empty. 5.57/2.20 Rewrite Strategy: FULL 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (3) DecreasingLoopProof (LOWER BOUND(ID)) 5.57/2.20 The following loop(s) give(s) rise to the lower bound Omega(n^1): 5.57/2.20 5.57/2.20 The rewrite sequence 5.57/2.20 5.57/2.20 activate(n__s(X)) ->^+ s(activate(X)) 5.57/2.20 5.57/2.20 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 5.57/2.20 5.57/2.20 The pumping substitution is [X / n__s(X)]. 5.57/2.20 5.57/2.20 The result substitution is [ ]. 5.57/2.20 5.57/2.20 5.57/2.20 5.57/2.20 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (4) 5.57/2.20 Complex Obligation (BEST) 5.57/2.20 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (5) 5.57/2.20 Obligation: 5.57/2.20 Proved the lower bound n^1 for the following obligation: 5.57/2.20 5.57/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.57/2.20 5.57/2.20 5.57/2.20 The TRS R consists of the following rules: 5.57/2.20 5.57/2.20 and(tt, T) -> T 5.57/2.20 isNatIList(IL) -> isNatList(activate(IL)) 5.57/2.20 isNat(n__0) -> tt 5.57/2.20 isNat(n__s(N)) -> isNat(activate(N)) 5.57/2.20 isNat(n__length(L)) -> isNatList(activate(L)) 5.57/2.20 isNatIList(n__zeros) -> tt 5.57/2.20 isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 isNatList(n__nil) -> tt 5.57/2.20 isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) 5.57/2.20 isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 zeros -> cons(0, n__zeros) 5.57/2.20 take(0, IL) -> uTake1(isNatIList(IL)) 5.57/2.20 uTake1(tt) -> nil 5.57/2.20 take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) 5.57/2.20 uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) 5.57/2.20 length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) 5.57/2.20 uLength(tt, L) -> s(length(activate(L))) 5.57/2.20 0 -> n__0 5.57/2.20 s(X) -> n__s(X) 5.57/2.20 length(X) -> n__length(X) 5.57/2.20 zeros -> n__zeros 5.57/2.20 cons(X1, X2) -> n__cons(X1, X2) 5.57/2.20 nil -> n__nil 5.57/2.20 take(X1, X2) -> n__take(X1, X2) 5.57/2.20 activate(n__0) -> 0 5.57/2.20 activate(n__s(X)) -> s(activate(X)) 5.57/2.20 activate(n__length(X)) -> length(activate(X)) 5.57/2.20 activate(n__zeros) -> zeros 5.57/2.20 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.57/2.20 activate(n__nil) -> nil 5.57/2.20 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 5.57/2.20 activate(X) -> X 5.57/2.20 5.57/2.20 S is empty. 5.57/2.20 Rewrite Strategy: FULL 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (6) LowerBoundPropagationProof (FINISHED) 5.57/2.20 Propagated lower bound. 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (7) 5.57/2.20 BOUNDS(n^1, INF) 5.57/2.20 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (8) 5.57/2.20 Obligation: 5.57/2.20 Analyzing the following TRS for decreasing loops: 5.57/2.20 5.57/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.57/2.20 5.57/2.20 5.57/2.20 The TRS R consists of the following rules: 5.57/2.20 5.57/2.20 and(tt, T) -> T 5.57/2.20 isNatIList(IL) -> isNatList(activate(IL)) 5.57/2.20 isNat(n__0) -> tt 5.57/2.20 isNat(n__s(N)) -> isNat(activate(N)) 5.57/2.20 isNat(n__length(L)) -> isNatList(activate(L)) 5.57/2.20 isNatIList(n__zeros) -> tt 5.57/2.20 isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 isNatList(n__nil) -> tt 5.57/2.20 isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) 5.57/2.20 isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) 5.57/2.20 zeros -> cons(0, n__zeros) 5.57/2.20 take(0, IL) -> uTake1(isNatIList(IL)) 5.57/2.20 uTake1(tt) -> nil 5.57/2.20 take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) 5.57/2.20 uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) 5.57/2.20 length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) 5.57/2.20 uLength(tt, L) -> s(length(activate(L))) 5.57/2.20 0 -> n__0 5.57/2.20 s(X) -> n__s(X) 5.57/2.20 length(X) -> n__length(X) 5.57/2.20 zeros -> n__zeros 5.57/2.20 cons(X1, X2) -> n__cons(X1, X2) 5.57/2.20 nil -> n__nil 5.57/2.20 take(X1, X2) -> n__take(X1, X2) 5.57/2.20 activate(n__0) -> 0 5.57/2.20 activate(n__s(X)) -> s(activate(X)) 5.57/2.20 activate(n__length(X)) -> length(activate(X)) 5.57/2.20 activate(n__zeros) -> zeros 5.57/2.20 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 5.57/2.20 activate(n__nil) -> nil 5.57/2.20 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 5.57/2.20 activate(X) -> X 5.57/2.20 5.57/2.20 S is empty. 5.57/2.20 Rewrite Strategy: FULL 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (9) DecreasingLoopProof (FINISHED) 5.57/2.20 The following loop(s) give(s) rise to the lower bound EXP: 5.57/2.20 5.57/2.20 The rewrite sequence 5.57/2.20 5.57/2.20 activate(n__length(n__cons(X11_0, X22_0))) ->^+ uLength(and(isNat(activate(X11_0)), isNatList(activate(X22_0))), activate(X22_0)) 5.57/2.20 5.57/2.20 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0]. 5.57/2.20 5.57/2.20 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 5.57/2.20 5.57/2.20 The result substitution is [ ]. 5.57/2.20 5.57/2.20 5.57/2.20 5.57/2.20 The rewrite sequence 5.57/2.20 5.57/2.20 activate(n__length(n__cons(X11_0, X22_0))) ->^+ uLength(and(isNat(activate(X11_0)), isNatList(activate(X22_0))), activate(X22_0)) 5.57/2.20 5.57/2.20 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 5.57/2.20 5.57/2.20 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 5.57/2.20 5.57/2.20 The result substitution is [ ]. 5.57/2.20 5.57/2.20 5.57/2.20 5.57/2.20 5.57/2.20 ---------------------------------------- 5.57/2.20 5.57/2.20 (10) 5.57/2.20 BOUNDS(EXP, INF) 5.62/2.23 EOF