322.24/291.49 WORST_CASE(Omega(n^1), ?) 322.24/291.50 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 322.24/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 322.24/291.50 322.24/291.50 322.24/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 322.24/291.50 322.24/291.50 (0) CpxTRS 322.24/291.50 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 322.24/291.50 (2) TRS for Loop Detection 322.24/291.50 (3) DecreasingLoopProof [LOWER BOUND(ID), 46 ms] 322.24/291.50 (4) BEST 322.24/291.50 (5) proven lower bound 322.24/291.50 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 322.24/291.50 (7) BOUNDS(n^1, INF) 322.24/291.50 (8) TRS for Loop Detection 322.24/291.50 322.24/291.50 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (0) 322.24/291.50 Obligation: 322.24/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 322.24/291.50 322.24/291.50 322.24/291.50 The TRS R consists of the following rules: 322.24/291.50 322.24/291.50 U11(tt, N) -> activate(N) 322.24/291.50 U21(tt, M, N) -> s(plus(activate(N), activate(M))) 322.24/291.50 U31(tt) -> 0 322.24/291.50 U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 322.24/291.50 and(tt, X) -> activate(X) 322.24/291.50 isNat(n__0) -> tt 322.24/291.50 isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 isNat(n__s(V1)) -> isNat(activate(V1)) 322.24/291.50 isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 plus(N, 0) -> U11(isNat(N), N) 322.24/291.50 plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 x(N, 0) -> U31(isNat(N)) 322.24/291.50 x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 0 -> n__0 322.24/291.50 plus(X1, X2) -> n__plus(X1, X2) 322.24/291.50 isNat(X) -> n__isNat(X) 322.24/291.50 s(X) -> n__s(X) 322.24/291.50 x(X1, X2) -> n__x(X1, X2) 322.24/291.50 activate(n__0) -> 0 322.24/291.50 activate(n__plus(X1, X2)) -> plus(X1, X2) 322.24/291.50 activate(n__isNat(X)) -> isNat(X) 322.24/291.50 activate(n__s(X)) -> s(X) 322.24/291.50 activate(n__x(X1, X2)) -> x(X1, X2) 322.24/291.50 activate(X) -> X 322.24/291.50 322.24/291.50 S is empty. 322.24/291.50 Rewrite Strategy: FULL 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 322.24/291.50 Transformed a relative TRS into a decreasing-loop problem. 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (2) 322.24/291.50 Obligation: 322.24/291.50 Analyzing the following TRS for decreasing loops: 322.24/291.50 322.24/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 322.24/291.50 322.24/291.50 322.24/291.50 The TRS R consists of the following rules: 322.24/291.50 322.24/291.50 U11(tt, N) -> activate(N) 322.24/291.50 U21(tt, M, N) -> s(plus(activate(N), activate(M))) 322.24/291.50 U31(tt) -> 0 322.24/291.50 U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 322.24/291.50 and(tt, X) -> activate(X) 322.24/291.50 isNat(n__0) -> tt 322.24/291.50 isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 isNat(n__s(V1)) -> isNat(activate(V1)) 322.24/291.50 isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 plus(N, 0) -> U11(isNat(N), N) 322.24/291.50 plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 x(N, 0) -> U31(isNat(N)) 322.24/291.50 x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 0 -> n__0 322.24/291.50 plus(X1, X2) -> n__plus(X1, X2) 322.24/291.50 isNat(X) -> n__isNat(X) 322.24/291.50 s(X) -> n__s(X) 322.24/291.50 x(X1, X2) -> n__x(X1, X2) 322.24/291.50 activate(n__0) -> 0 322.24/291.50 activate(n__plus(X1, X2)) -> plus(X1, X2) 322.24/291.50 activate(n__isNat(X)) -> isNat(X) 322.24/291.50 activate(n__s(X)) -> s(X) 322.24/291.50 activate(n__x(X1, X2)) -> x(X1, X2) 322.24/291.50 activate(X) -> X 322.24/291.50 322.24/291.50 S is empty. 322.24/291.50 Rewrite Strategy: FULL 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (3) DecreasingLoopProof (LOWER BOUND(ID)) 322.24/291.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 322.24/291.50 322.24/291.50 The rewrite sequence 322.24/291.50 322.24/291.50 isNat(n__plus(V1, V2)) ->^+ and(isNat(V1), n__isNat(activate(V2))) 322.24/291.50 322.24/291.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 322.24/291.50 322.24/291.50 The pumping substitution is [V1 / n__plus(V1, V2)]. 322.24/291.50 322.24/291.50 The result substitution is [ ]. 322.24/291.50 322.24/291.50 322.24/291.50 322.24/291.50 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (4) 322.24/291.50 Complex Obligation (BEST) 322.24/291.50 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (5) 322.24/291.50 Obligation: 322.24/291.50 Proved the lower bound n^1 for the following obligation: 322.24/291.50 322.24/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 322.24/291.50 322.24/291.50 322.24/291.50 The TRS R consists of the following rules: 322.24/291.50 322.24/291.50 U11(tt, N) -> activate(N) 322.24/291.50 U21(tt, M, N) -> s(plus(activate(N), activate(M))) 322.24/291.50 U31(tt) -> 0 322.24/291.50 U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 322.24/291.50 and(tt, X) -> activate(X) 322.24/291.50 isNat(n__0) -> tt 322.24/291.50 isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 isNat(n__s(V1)) -> isNat(activate(V1)) 322.24/291.50 isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 plus(N, 0) -> U11(isNat(N), N) 322.24/291.50 plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 x(N, 0) -> U31(isNat(N)) 322.24/291.50 x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 0 -> n__0 322.24/291.50 plus(X1, X2) -> n__plus(X1, X2) 322.24/291.50 isNat(X) -> n__isNat(X) 322.24/291.50 s(X) -> n__s(X) 322.24/291.50 x(X1, X2) -> n__x(X1, X2) 322.24/291.50 activate(n__0) -> 0 322.24/291.50 activate(n__plus(X1, X2)) -> plus(X1, X2) 322.24/291.50 activate(n__isNat(X)) -> isNat(X) 322.24/291.50 activate(n__s(X)) -> s(X) 322.24/291.50 activate(n__x(X1, X2)) -> x(X1, X2) 322.24/291.50 activate(X) -> X 322.24/291.50 322.24/291.50 S is empty. 322.24/291.50 Rewrite Strategy: FULL 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (6) LowerBoundPropagationProof (FINISHED) 322.24/291.50 Propagated lower bound. 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (7) 322.24/291.50 BOUNDS(n^1, INF) 322.24/291.50 322.24/291.50 ---------------------------------------- 322.24/291.50 322.24/291.50 (8) 322.24/291.50 Obligation: 322.24/291.50 Analyzing the following TRS for decreasing loops: 322.24/291.50 322.24/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 322.24/291.50 322.24/291.50 322.24/291.50 The TRS R consists of the following rules: 322.24/291.50 322.24/291.50 U11(tt, N) -> activate(N) 322.24/291.50 U21(tt, M, N) -> s(plus(activate(N), activate(M))) 322.24/291.50 U31(tt) -> 0 322.24/291.50 U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 322.24/291.50 and(tt, X) -> activate(X) 322.24/291.50 isNat(n__0) -> tt 322.24/291.50 isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 isNat(n__s(V1)) -> isNat(activate(V1)) 322.24/291.50 isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) 322.24/291.50 plus(N, 0) -> U11(isNat(N), N) 322.24/291.50 plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 x(N, 0) -> U31(isNat(N)) 322.24/291.50 x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 322.24/291.50 0 -> n__0 322.24/291.50 plus(X1, X2) -> n__plus(X1, X2) 322.24/291.50 isNat(X) -> n__isNat(X) 322.24/291.50 s(X) -> n__s(X) 322.24/291.50 x(X1, X2) -> n__x(X1, X2) 322.24/291.50 activate(n__0) -> 0 322.24/291.50 activate(n__plus(X1, X2)) -> plus(X1, X2) 322.24/291.50 activate(n__isNat(X)) -> isNat(X) 322.24/291.50 activate(n__s(X)) -> s(X) 322.24/291.50 activate(n__x(X1, X2)) -> x(X1, X2) 322.24/291.50 activate(X) -> X 322.24/291.50 322.24/291.50 S is empty. 322.24/291.50 Rewrite Strategy: FULL 322.32/291.53 EOF