318.30/291.53 WORST_CASE(Omega(n^1), ?) 318.32/291.54 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 318.32/291.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 318.32/291.54 318.32/291.54 318.32/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.32/291.54 318.32/291.54 (0) CpxTRS 318.32/291.54 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 318.32/291.54 (2) TRS for Loop Detection 318.32/291.54 (3) DecreasingLoopProof [LOWER BOUND(ID), 93 ms] 318.32/291.54 (4) BEST 318.32/291.54 (5) proven lower bound 318.32/291.54 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 318.32/291.54 (7) BOUNDS(n^1, INF) 318.32/291.54 (8) TRS for Loop Detection 318.32/291.54 318.32/291.54 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (0) 318.32/291.54 Obligation: 318.32/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.32/291.54 318.32/291.54 318.32/291.54 The TRS R consists of the following rules: 318.32/291.54 318.32/291.54 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 318.32/291.54 U12(tt, V2) -> U13(isNat(activate(V2))) 318.32/291.54 U13(tt) -> tt 318.32/291.54 U21(tt, V1) -> U22(isNat(activate(V1))) 318.32/291.54 U22(tt) -> tt 318.32/291.54 U31(tt, N) -> activate(N) 318.32/291.54 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 318.32/291.54 and(tt, X) -> activate(X) 318.32/291.54 isNat(n__0) -> tt 318.32/291.54 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 318.32/291.54 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 318.32/291.54 isNatKind(n__0) -> tt 318.32/291.54 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 318.32/291.54 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 318.32/291.54 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 318.32/291.54 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 318.32/291.54 0 -> n__0 318.32/291.54 plus(X1, X2) -> n__plus(X1, X2) 318.32/291.54 isNatKind(X) -> n__isNatKind(X) 318.32/291.54 s(X) -> n__s(X) 318.32/291.54 and(X1, X2) -> n__and(X1, X2) 318.32/291.54 activate(n__0) -> 0 318.32/291.54 activate(n__plus(X1, X2)) -> plus(X1, X2) 318.32/291.54 activate(n__isNatKind(X)) -> isNatKind(X) 318.32/291.54 activate(n__s(X)) -> s(X) 318.32/291.54 activate(n__and(X1, X2)) -> and(X1, X2) 318.32/291.54 activate(X) -> X 318.32/291.54 318.32/291.54 S is empty. 318.32/291.54 Rewrite Strategy: FULL 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 318.32/291.54 Transformed a relative TRS into a decreasing-loop problem. 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (2) 318.32/291.54 Obligation: 318.32/291.54 Analyzing the following TRS for decreasing loops: 318.32/291.54 318.32/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.32/291.54 318.32/291.54 318.32/291.54 The TRS R consists of the following rules: 318.32/291.54 318.32/291.54 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 318.32/291.54 U12(tt, V2) -> U13(isNat(activate(V2))) 318.32/291.54 U13(tt) -> tt 318.32/291.54 U21(tt, V1) -> U22(isNat(activate(V1))) 318.32/291.54 U22(tt) -> tt 318.32/291.54 U31(tt, N) -> activate(N) 318.32/291.54 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 318.32/291.54 and(tt, X) -> activate(X) 318.32/291.54 isNat(n__0) -> tt 318.32/291.54 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 318.32/291.54 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 318.32/291.54 isNatKind(n__0) -> tt 318.32/291.54 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 318.32/291.54 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 318.32/291.54 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 318.32/291.54 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 318.32/291.54 0 -> n__0 318.32/291.54 plus(X1, X2) -> n__plus(X1, X2) 318.32/291.54 isNatKind(X) -> n__isNatKind(X) 318.32/291.54 s(X) -> n__s(X) 318.32/291.54 and(X1, X2) -> n__and(X1, X2) 318.32/291.54 activate(n__0) -> 0 318.32/291.54 activate(n__plus(X1, X2)) -> plus(X1, X2) 318.32/291.54 activate(n__isNatKind(X)) -> isNatKind(X) 318.32/291.54 activate(n__s(X)) -> s(X) 318.32/291.54 activate(n__and(X1, X2)) -> and(X1, X2) 318.32/291.54 activate(X) -> X 318.32/291.54 318.32/291.54 S is empty. 318.32/291.54 Rewrite Strategy: FULL 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (3) DecreasingLoopProof (LOWER BOUND(ID)) 318.32/291.54 The following loop(s) give(s) rise to the lower bound Omega(n^1): 318.32/291.54 318.32/291.54 The rewrite sequence 318.32/291.54 318.32/291.54 isNatKind(n__plus(V1, V2)) ->^+ and(isNatKind(V1), n__isNatKind(activate(V2))) 318.32/291.54 318.32/291.54 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 318.32/291.54 318.32/291.54 The pumping substitution is [V1 / n__plus(V1, V2)]. 318.32/291.54 318.32/291.54 The result substitution is [ ]. 318.32/291.54 318.32/291.54 318.32/291.54 318.32/291.54 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (4) 318.32/291.54 Complex Obligation (BEST) 318.32/291.54 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (5) 318.32/291.54 Obligation: 318.32/291.54 Proved the lower bound n^1 for the following obligation: 318.32/291.54 318.32/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.32/291.54 318.32/291.54 318.32/291.54 The TRS R consists of the following rules: 318.32/291.54 318.32/291.54 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 318.32/291.54 U12(tt, V2) -> U13(isNat(activate(V2))) 318.32/291.54 U13(tt) -> tt 318.32/291.54 U21(tt, V1) -> U22(isNat(activate(V1))) 318.32/291.54 U22(tt) -> tt 318.32/291.54 U31(tt, N) -> activate(N) 318.32/291.54 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 318.32/291.54 and(tt, X) -> activate(X) 318.32/291.54 isNat(n__0) -> tt 318.32/291.54 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 318.32/291.54 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 318.32/291.54 isNatKind(n__0) -> tt 318.32/291.54 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 318.32/291.54 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 318.32/291.54 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 318.32/291.54 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 318.32/291.54 0 -> n__0 318.32/291.54 plus(X1, X2) -> n__plus(X1, X2) 318.32/291.54 isNatKind(X) -> n__isNatKind(X) 318.32/291.54 s(X) -> n__s(X) 318.32/291.54 and(X1, X2) -> n__and(X1, X2) 318.32/291.54 activate(n__0) -> 0 318.32/291.54 activate(n__plus(X1, X2)) -> plus(X1, X2) 318.32/291.54 activate(n__isNatKind(X)) -> isNatKind(X) 318.32/291.54 activate(n__s(X)) -> s(X) 318.32/291.54 activate(n__and(X1, X2)) -> and(X1, X2) 318.32/291.54 activate(X) -> X 318.32/291.54 318.32/291.54 S is empty. 318.32/291.54 Rewrite Strategy: FULL 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (6) LowerBoundPropagationProof (FINISHED) 318.32/291.54 Propagated lower bound. 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (7) 318.32/291.54 BOUNDS(n^1, INF) 318.32/291.54 318.32/291.54 ---------------------------------------- 318.32/291.54 318.32/291.54 (8) 318.32/291.54 Obligation: 318.32/291.54 Analyzing the following TRS for decreasing loops: 318.32/291.54 318.32/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.32/291.54 318.32/291.54 318.32/291.54 The TRS R consists of the following rules: 318.32/291.54 318.32/291.54 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 318.32/291.54 U12(tt, V2) -> U13(isNat(activate(V2))) 318.32/291.54 U13(tt) -> tt 318.32/291.54 U21(tt, V1) -> U22(isNat(activate(V1))) 318.32/291.54 U22(tt) -> tt 318.32/291.54 U31(tt, N) -> activate(N) 318.32/291.54 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 318.32/291.54 and(tt, X) -> activate(X) 318.32/291.54 isNat(n__0) -> tt 318.32/291.54 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 318.32/291.54 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 318.32/291.54 isNatKind(n__0) -> tt 318.32/291.54 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 318.32/291.54 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 318.32/291.54 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 318.32/291.54 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 318.32/291.54 0 -> n__0 318.32/291.54 plus(X1, X2) -> n__plus(X1, X2) 318.32/291.54 isNatKind(X) -> n__isNatKind(X) 318.32/291.54 s(X) -> n__s(X) 318.32/291.54 and(X1, X2) -> n__and(X1, X2) 318.32/291.54 activate(n__0) -> 0 318.32/291.54 activate(n__plus(X1, X2)) -> plus(X1, X2) 318.32/291.54 activate(n__isNatKind(X)) -> isNatKind(X) 318.32/291.54 activate(n__s(X)) -> s(X) 318.32/291.54 activate(n__and(X1, X2)) -> and(X1, X2) 318.32/291.54 activate(X) -> X 318.32/291.54 318.32/291.54 S is empty. 318.32/291.54 Rewrite Strategy: FULL 318.32/291.57 EOF