4.27/1.89 WORST_CASE(NON_POLY, ?) 4.27/1.90 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.27/1.90 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.27/1.90 4.27/1.90 4.27/1.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.27/1.90 4.27/1.90 (0) CpxTRS 4.27/1.90 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 4.27/1.90 (2) TRS for Loop Detection 4.27/1.90 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 4.27/1.90 (4) BEST 4.27/1.90 (5) proven lower bound 4.27/1.90 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 4.27/1.90 (7) BOUNDS(n^1, INF) 4.27/1.90 (8) TRS for Loop Detection 4.27/1.90 (9) DecreasingLoopProof [FINISHED, 82 ms] 4.27/1.90 (10) BOUNDS(EXP, INF) 4.27/1.90 4.27/1.90 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (0) 4.27/1.90 Obligation: 4.27/1.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.27/1.90 4.27/1.90 4.27/1.90 The TRS R consists of the following rules: 4.27/1.90 4.27/1.90 zeros -> cons(0, n__zeros) 4.27/1.90 U11(tt, V1) -> U12(isNatList(activate(V1))) 4.27/1.90 U12(tt) -> tt 4.27/1.90 U21(tt, V1) -> U22(isNat(activate(V1))) 4.27/1.90 U22(tt) -> tt 4.27/1.90 U31(tt, V) -> U32(isNatList(activate(V))) 4.27/1.90 U32(tt) -> tt 4.27/1.90 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 4.27/1.90 U42(tt, V2) -> U43(isNatIList(activate(V2))) 4.27/1.90 U43(tt) -> tt 4.27/1.90 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 4.27/1.90 U52(tt, V2) -> U53(isNatList(activate(V2))) 4.27/1.90 U53(tt) -> tt 4.27/1.90 U61(tt, L) -> s(length(activate(L))) 4.27/1.90 and(tt, X) -> activate(X) 4.27/1.90 isNat(n__0) -> tt 4.27/1.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 4.27/1.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 4.27/1.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 4.27/1.90 isNatIList(n__zeros) -> tt 4.27/1.90 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 isNatIListKind(n__nil) -> tt 4.27/1.90 isNatIListKind(n__zeros) -> tt 4.27/1.90 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 4.27/1.90 isNatKind(n__0) -> tt 4.27/1.90 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 4.27/1.90 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 4.27/1.90 isNatList(n__nil) -> tt 4.27/1.90 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 length(nil) -> 0 4.27/1.90 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L)) 4.27/1.90 zeros -> n__zeros 4.27/1.90 0 -> n__0 4.27/1.90 length(X) -> n__length(X) 4.27/1.90 s(X) -> n__s(X) 4.27/1.90 cons(X1, X2) -> n__cons(X1, X2) 4.27/1.90 isNatIListKind(X) -> n__isNatIListKind(X) 4.27/1.90 nil -> n__nil 4.27/1.90 and(X1, X2) -> n__and(X1, X2) 4.27/1.90 isNat(X) -> n__isNat(X) 4.27/1.90 isNatKind(X) -> n__isNatKind(X) 4.27/1.90 activate(n__zeros) -> zeros 4.27/1.90 activate(n__0) -> 0 4.27/1.90 activate(n__length(X)) -> length(activate(X)) 4.27/1.90 activate(n__s(X)) -> s(activate(X)) 4.27/1.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 4.27/1.90 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 4.27/1.90 activate(n__nil) -> nil 4.27/1.90 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.27/1.90 activate(n__isNat(X)) -> isNat(X) 4.27/1.90 activate(n__isNatKind(X)) -> isNatKind(X) 4.27/1.90 activate(X) -> X 4.27/1.90 4.27/1.90 S is empty. 4.27/1.90 Rewrite Strategy: FULL 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 4.27/1.90 Transformed a relative TRS into a decreasing-loop problem. 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (2) 4.27/1.90 Obligation: 4.27/1.90 Analyzing the following TRS for decreasing loops: 4.27/1.90 4.27/1.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.27/1.90 4.27/1.90 4.27/1.90 The TRS R consists of the following rules: 4.27/1.90 4.27/1.90 zeros -> cons(0, n__zeros) 4.27/1.90 U11(tt, V1) -> U12(isNatList(activate(V1))) 4.27/1.90 U12(tt) -> tt 4.27/1.90 U21(tt, V1) -> U22(isNat(activate(V1))) 4.27/1.90 U22(tt) -> tt 4.27/1.90 U31(tt, V) -> U32(isNatList(activate(V))) 4.27/1.90 U32(tt) -> tt 4.27/1.90 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 4.27/1.90 U42(tt, V2) -> U43(isNatIList(activate(V2))) 4.27/1.90 U43(tt) -> tt 4.27/1.90 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 4.27/1.90 U52(tt, V2) -> U53(isNatList(activate(V2))) 4.27/1.90 U53(tt) -> tt 4.27/1.90 U61(tt, L) -> s(length(activate(L))) 4.27/1.90 and(tt, X) -> activate(X) 4.27/1.90 isNat(n__0) -> tt 4.27/1.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 4.27/1.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 4.27/1.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 4.27/1.90 isNatIList(n__zeros) -> tt 4.27/1.90 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 isNatIListKind(n__nil) -> tt 4.27/1.90 isNatIListKind(n__zeros) -> tt 4.27/1.90 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 4.27/1.90 isNatKind(n__0) -> tt 4.27/1.90 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 4.27/1.90 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 4.27/1.90 isNatList(n__nil) -> tt 4.27/1.90 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 length(nil) -> 0 4.27/1.90 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L)) 4.27/1.90 zeros -> n__zeros 4.27/1.90 0 -> n__0 4.27/1.90 length(X) -> n__length(X) 4.27/1.90 s(X) -> n__s(X) 4.27/1.90 cons(X1, X2) -> n__cons(X1, X2) 4.27/1.90 isNatIListKind(X) -> n__isNatIListKind(X) 4.27/1.90 nil -> n__nil 4.27/1.90 and(X1, X2) -> n__and(X1, X2) 4.27/1.90 isNat(X) -> n__isNat(X) 4.27/1.90 isNatKind(X) -> n__isNatKind(X) 4.27/1.90 activate(n__zeros) -> zeros 4.27/1.90 activate(n__0) -> 0 4.27/1.90 activate(n__length(X)) -> length(activate(X)) 4.27/1.90 activate(n__s(X)) -> s(activate(X)) 4.27/1.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 4.27/1.90 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 4.27/1.90 activate(n__nil) -> nil 4.27/1.90 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.27/1.90 activate(n__isNat(X)) -> isNat(X) 4.27/1.90 activate(n__isNatKind(X)) -> isNatKind(X) 4.27/1.90 activate(X) -> X 4.27/1.90 4.27/1.90 S is empty. 4.27/1.90 Rewrite Strategy: FULL 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (3) DecreasingLoopProof (LOWER BOUND(ID)) 4.27/1.90 The following loop(s) give(s) rise to the lower bound Omega(n^1): 4.27/1.90 4.27/1.90 The rewrite sequence 4.27/1.90 4.27/1.90 activate(n__s(X)) ->^+ s(activate(X)) 4.27/1.90 4.27/1.90 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 4.27/1.90 4.27/1.90 The pumping substitution is [X / n__s(X)]. 4.27/1.90 4.27/1.90 The result substitution is [ ]. 4.27/1.90 4.27/1.90 4.27/1.90 4.27/1.90 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (4) 4.27/1.90 Complex Obligation (BEST) 4.27/1.90 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (5) 4.27/1.90 Obligation: 4.27/1.90 Proved the lower bound n^1 for the following obligation: 4.27/1.90 4.27/1.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.27/1.90 4.27/1.90 4.27/1.90 The TRS R consists of the following rules: 4.27/1.90 4.27/1.90 zeros -> cons(0, n__zeros) 4.27/1.90 U11(tt, V1) -> U12(isNatList(activate(V1))) 4.27/1.90 U12(tt) -> tt 4.27/1.90 U21(tt, V1) -> U22(isNat(activate(V1))) 4.27/1.90 U22(tt) -> tt 4.27/1.90 U31(tt, V) -> U32(isNatList(activate(V))) 4.27/1.90 U32(tt) -> tt 4.27/1.90 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 4.27/1.90 U42(tt, V2) -> U43(isNatIList(activate(V2))) 4.27/1.90 U43(tt) -> tt 4.27/1.90 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 4.27/1.90 U52(tt, V2) -> U53(isNatList(activate(V2))) 4.27/1.90 U53(tt) -> tt 4.27/1.90 U61(tt, L) -> s(length(activate(L))) 4.27/1.90 and(tt, X) -> activate(X) 4.27/1.90 isNat(n__0) -> tt 4.27/1.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 4.27/1.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 4.27/1.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 4.27/1.90 isNatIList(n__zeros) -> tt 4.27/1.90 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 isNatIListKind(n__nil) -> tt 4.27/1.90 isNatIListKind(n__zeros) -> tt 4.27/1.90 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 4.27/1.90 isNatKind(n__0) -> tt 4.27/1.90 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 4.27/1.90 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 4.27/1.90 isNatList(n__nil) -> tt 4.27/1.90 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 length(nil) -> 0 4.27/1.90 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L)) 4.27/1.90 zeros -> n__zeros 4.27/1.90 0 -> n__0 4.27/1.90 length(X) -> n__length(X) 4.27/1.90 s(X) -> n__s(X) 4.27/1.90 cons(X1, X2) -> n__cons(X1, X2) 4.27/1.90 isNatIListKind(X) -> n__isNatIListKind(X) 4.27/1.90 nil -> n__nil 4.27/1.90 and(X1, X2) -> n__and(X1, X2) 4.27/1.90 isNat(X) -> n__isNat(X) 4.27/1.90 isNatKind(X) -> n__isNatKind(X) 4.27/1.90 activate(n__zeros) -> zeros 4.27/1.90 activate(n__0) -> 0 4.27/1.90 activate(n__length(X)) -> length(activate(X)) 4.27/1.90 activate(n__s(X)) -> s(activate(X)) 4.27/1.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 4.27/1.90 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 4.27/1.90 activate(n__nil) -> nil 4.27/1.90 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.27/1.90 activate(n__isNat(X)) -> isNat(X) 4.27/1.90 activate(n__isNatKind(X)) -> isNatKind(X) 4.27/1.90 activate(X) -> X 4.27/1.90 4.27/1.90 S is empty. 4.27/1.90 Rewrite Strategy: FULL 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (6) LowerBoundPropagationProof (FINISHED) 4.27/1.90 Propagated lower bound. 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (7) 4.27/1.90 BOUNDS(n^1, INF) 4.27/1.90 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (8) 4.27/1.90 Obligation: 4.27/1.90 Analyzing the following TRS for decreasing loops: 4.27/1.90 4.27/1.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.27/1.90 4.27/1.90 4.27/1.90 The TRS R consists of the following rules: 4.27/1.90 4.27/1.90 zeros -> cons(0, n__zeros) 4.27/1.90 U11(tt, V1) -> U12(isNatList(activate(V1))) 4.27/1.90 U12(tt) -> tt 4.27/1.90 U21(tt, V1) -> U22(isNat(activate(V1))) 4.27/1.90 U22(tt) -> tt 4.27/1.90 U31(tt, V) -> U32(isNatList(activate(V))) 4.27/1.90 U32(tt) -> tt 4.27/1.90 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 4.27/1.90 U42(tt, V2) -> U43(isNatIList(activate(V2))) 4.27/1.90 U43(tt) -> tt 4.27/1.90 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 4.27/1.90 U52(tt, V2) -> U53(isNatList(activate(V2))) 4.27/1.90 U53(tt) -> tt 4.27/1.90 U61(tt, L) -> s(length(activate(L))) 4.27/1.90 and(tt, X) -> activate(X) 4.27/1.90 isNat(n__0) -> tt 4.27/1.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 4.27/1.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 4.27/1.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 4.27/1.90 isNatIList(n__zeros) -> tt 4.27/1.90 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 isNatIListKind(n__nil) -> tt 4.27/1.90 isNatIListKind(n__zeros) -> tt 4.27/1.90 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 4.27/1.90 isNatKind(n__0) -> tt 4.27/1.90 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 4.27/1.90 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 4.27/1.90 isNatList(n__nil) -> tt 4.27/1.90 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 4.27/1.90 length(nil) -> 0 4.27/1.90 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L)) 4.27/1.90 zeros -> n__zeros 4.27/1.90 0 -> n__0 4.27/1.90 length(X) -> n__length(X) 4.27/1.90 s(X) -> n__s(X) 4.27/1.90 cons(X1, X2) -> n__cons(X1, X2) 4.27/1.90 isNatIListKind(X) -> n__isNatIListKind(X) 4.27/1.90 nil -> n__nil 4.27/1.90 and(X1, X2) -> n__and(X1, X2) 4.27/1.90 isNat(X) -> n__isNat(X) 4.27/1.90 isNatKind(X) -> n__isNatKind(X) 4.27/1.90 activate(n__zeros) -> zeros 4.27/1.90 activate(n__0) -> 0 4.27/1.90 activate(n__length(X)) -> length(activate(X)) 4.27/1.90 activate(n__s(X)) -> s(activate(X)) 4.27/1.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 4.27/1.90 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 4.27/1.90 activate(n__nil) -> nil 4.27/1.90 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.27/1.90 activate(n__isNat(X)) -> isNat(X) 4.27/1.90 activate(n__isNatKind(X)) -> isNatKind(X) 4.27/1.90 activate(X) -> X 4.27/1.90 4.27/1.90 S is empty. 4.27/1.90 Rewrite Strategy: FULL 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (9) DecreasingLoopProof (FINISHED) 4.27/1.90 The following loop(s) give(s) rise to the lower bound EXP: 4.27/1.90 4.27/1.90 The rewrite sequence 4.27/1.90 4.27/1.90 activate(n__isNat(n__s(V11_0))) ->^+ U21(isNatKind(activate(V11_0)), activate(V11_0)) 4.27/1.90 4.27/1.90 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 4.27/1.90 4.27/1.90 The pumping substitution is [V11_0 / n__isNat(n__s(V11_0))]. 4.27/1.90 4.27/1.90 The result substitution is [ ]. 4.27/1.90 4.27/1.90 4.27/1.90 4.27/1.90 The rewrite sequence 4.27/1.90 4.27/1.90 activate(n__isNat(n__s(V11_0))) ->^+ U21(isNatKind(activate(V11_0)), activate(V11_0)) 4.27/1.90 4.27/1.90 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 4.27/1.90 4.27/1.90 The pumping substitution is [V11_0 / n__isNat(n__s(V11_0))]. 4.27/1.90 4.27/1.90 The result substitution is [ ]. 4.27/1.90 4.27/1.90 4.27/1.90 4.27/1.90 4.27/1.90 ---------------------------------------- 4.27/1.90 4.27/1.90 (10) 4.27/1.90 BOUNDS(EXP, INF) 4.36/1.94 EOF