316.83/292.16 WORST_CASE(Omega(n^1), ?) 316.83/292.17 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 316.83/292.17 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 316.83/292.17 316.83/292.17 316.83/292.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.83/292.17 316.83/292.17 (0) CpxTRS 316.83/292.17 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 316.83/292.17 (2) TRS for Loop Detection 316.83/292.17 (3) DecreasingLoopProof [LOWER BOUND(ID), 124 ms] 316.83/292.17 (4) BEST 316.83/292.17 (5) proven lower bound 316.83/292.17 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 316.83/292.17 (7) BOUNDS(n^1, INF) 316.83/292.17 (8) TRS for Loop Detection 316.83/292.17 316.83/292.17 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (0) 316.83/292.17 Obligation: 316.83/292.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.83/292.17 316.83/292.17 316.83/292.17 The TRS R consists of the following rules: 316.83/292.17 316.83/292.17 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 316.83/292.17 U12(tt, V2) -> U13(isNat(activate(V2))) 316.83/292.17 U13(tt) -> tt 316.83/292.17 U21(tt, V1) -> U22(isNat(activate(V1))) 316.83/292.17 U22(tt) -> tt 316.83/292.17 U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) 316.83/292.17 U32(tt, V2) -> U33(isNat(activate(V2))) 316.83/292.17 U33(tt) -> tt 316.83/292.17 U41(tt, N) -> activate(N) 316.83/292.17 U51(tt, M, N) -> s(plus(activate(N), activate(M))) 316.83/292.17 U61(tt) -> 0 316.83/292.17 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 316.83/292.17 and(tt, X) -> activate(X) 316.83/292.17 isNat(n__0) -> tt 316.83/292.17 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 316.83/292.17 isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNatKind(n__0) -> tt 316.83/292.17 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 316.83/292.17 isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) 316.83/292.17 plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) 316.83/292.17 x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 0 -> n__0 316.83/292.17 plus(X1, X2) -> n__plus(X1, X2) 316.83/292.17 isNatKind(X) -> n__isNatKind(X) 316.83/292.17 s(X) -> n__s(X) 316.83/292.17 x(X1, X2) -> n__x(X1, X2) 316.83/292.17 and(X1, X2) -> n__and(X1, X2) 316.83/292.17 activate(n__0) -> 0 316.83/292.17 activate(n__plus(X1, X2)) -> plus(X1, X2) 316.83/292.17 activate(n__isNatKind(X)) -> isNatKind(X) 316.83/292.17 activate(n__s(X)) -> s(X) 316.83/292.17 activate(n__x(X1, X2)) -> x(X1, X2) 316.83/292.17 activate(n__and(X1, X2)) -> and(X1, X2) 316.83/292.17 activate(X) -> X 316.83/292.17 316.83/292.17 S is empty. 316.83/292.17 Rewrite Strategy: FULL 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 316.83/292.17 Transformed a relative TRS into a decreasing-loop problem. 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (2) 316.83/292.17 Obligation: 316.83/292.17 Analyzing the following TRS for decreasing loops: 316.83/292.17 316.83/292.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.83/292.17 316.83/292.17 316.83/292.17 The TRS R consists of the following rules: 316.83/292.17 316.83/292.17 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 316.83/292.17 U12(tt, V2) -> U13(isNat(activate(V2))) 316.83/292.17 U13(tt) -> tt 316.83/292.17 U21(tt, V1) -> U22(isNat(activate(V1))) 316.83/292.17 U22(tt) -> tt 316.83/292.17 U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) 316.83/292.17 U32(tt, V2) -> U33(isNat(activate(V2))) 316.83/292.17 U33(tt) -> tt 316.83/292.17 U41(tt, N) -> activate(N) 316.83/292.17 U51(tt, M, N) -> s(plus(activate(N), activate(M))) 316.83/292.17 U61(tt) -> 0 316.83/292.17 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 316.83/292.17 and(tt, X) -> activate(X) 316.83/292.17 isNat(n__0) -> tt 316.83/292.17 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 316.83/292.17 isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNatKind(n__0) -> tt 316.83/292.17 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 316.83/292.17 isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) 316.83/292.17 plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) 316.83/292.17 x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 0 -> n__0 316.83/292.17 plus(X1, X2) -> n__plus(X1, X2) 316.83/292.17 isNatKind(X) -> n__isNatKind(X) 316.83/292.17 s(X) -> n__s(X) 316.83/292.17 x(X1, X2) -> n__x(X1, X2) 316.83/292.17 and(X1, X2) -> n__and(X1, X2) 316.83/292.17 activate(n__0) -> 0 316.83/292.17 activate(n__plus(X1, X2)) -> plus(X1, X2) 316.83/292.17 activate(n__isNatKind(X)) -> isNatKind(X) 316.83/292.17 activate(n__s(X)) -> s(X) 316.83/292.17 activate(n__x(X1, X2)) -> x(X1, X2) 316.83/292.17 activate(n__and(X1, X2)) -> and(X1, X2) 316.83/292.17 activate(X) -> X 316.83/292.17 316.83/292.17 S is empty. 316.83/292.17 Rewrite Strategy: FULL 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (3) DecreasingLoopProof (LOWER BOUND(ID)) 316.83/292.17 The following loop(s) give(s) rise to the lower bound Omega(n^1): 316.83/292.17 316.83/292.17 The rewrite sequence 316.83/292.17 316.83/292.17 isNatKind(n__plus(V1, V2)) ->^+ and(isNatKind(V1), n__isNatKind(activate(V2))) 316.83/292.17 316.83/292.17 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 316.83/292.17 316.83/292.17 The pumping substitution is [V1 / n__plus(V1, V2)]. 316.83/292.17 316.83/292.17 The result substitution is [ ]. 316.83/292.17 316.83/292.17 316.83/292.17 316.83/292.17 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (4) 316.83/292.17 Complex Obligation (BEST) 316.83/292.17 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (5) 316.83/292.17 Obligation: 316.83/292.17 Proved the lower bound n^1 for the following obligation: 316.83/292.17 316.83/292.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.83/292.17 316.83/292.17 316.83/292.17 The TRS R consists of the following rules: 316.83/292.17 316.83/292.17 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 316.83/292.17 U12(tt, V2) -> U13(isNat(activate(V2))) 316.83/292.17 U13(tt) -> tt 316.83/292.17 U21(tt, V1) -> U22(isNat(activate(V1))) 316.83/292.17 U22(tt) -> tt 316.83/292.17 U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) 316.83/292.17 U32(tt, V2) -> U33(isNat(activate(V2))) 316.83/292.17 U33(tt) -> tt 316.83/292.17 U41(tt, N) -> activate(N) 316.83/292.17 U51(tt, M, N) -> s(plus(activate(N), activate(M))) 316.83/292.17 U61(tt) -> 0 316.83/292.17 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 316.83/292.17 and(tt, X) -> activate(X) 316.83/292.17 isNat(n__0) -> tt 316.83/292.17 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 316.83/292.17 isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNatKind(n__0) -> tt 316.83/292.17 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 316.83/292.17 isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) 316.83/292.17 plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) 316.83/292.17 x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 0 -> n__0 316.83/292.17 plus(X1, X2) -> n__plus(X1, X2) 316.83/292.17 isNatKind(X) -> n__isNatKind(X) 316.83/292.17 s(X) -> n__s(X) 316.83/292.17 x(X1, X2) -> n__x(X1, X2) 316.83/292.17 and(X1, X2) -> n__and(X1, X2) 316.83/292.17 activate(n__0) -> 0 316.83/292.17 activate(n__plus(X1, X2)) -> plus(X1, X2) 316.83/292.17 activate(n__isNatKind(X)) -> isNatKind(X) 316.83/292.17 activate(n__s(X)) -> s(X) 316.83/292.17 activate(n__x(X1, X2)) -> x(X1, X2) 316.83/292.17 activate(n__and(X1, X2)) -> and(X1, X2) 316.83/292.17 activate(X) -> X 316.83/292.17 316.83/292.17 S is empty. 316.83/292.17 Rewrite Strategy: FULL 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (6) LowerBoundPropagationProof (FINISHED) 316.83/292.17 Propagated lower bound. 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (7) 316.83/292.17 BOUNDS(n^1, INF) 316.83/292.17 316.83/292.17 ---------------------------------------- 316.83/292.17 316.83/292.17 (8) 316.83/292.17 Obligation: 316.83/292.17 Analyzing the following TRS for decreasing loops: 316.83/292.17 316.83/292.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.83/292.17 316.83/292.17 316.83/292.17 The TRS R consists of the following rules: 316.83/292.17 316.83/292.17 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 316.83/292.17 U12(tt, V2) -> U13(isNat(activate(V2))) 316.83/292.17 U13(tt) -> tt 316.83/292.17 U21(tt, V1) -> U22(isNat(activate(V1))) 316.83/292.17 U22(tt) -> tt 316.83/292.17 U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) 316.83/292.17 U32(tt, V2) -> U33(isNat(activate(V2))) 316.83/292.17 U33(tt) -> tt 316.83/292.17 U41(tt, N) -> activate(N) 316.83/292.17 U51(tt, M, N) -> s(plus(activate(N), activate(M))) 316.83/292.17 U61(tt) -> 0 316.83/292.17 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 316.83/292.17 and(tt, X) -> activate(X) 316.83/292.17 isNat(n__0) -> tt 316.83/292.17 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 316.83/292.17 isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 316.83/292.17 isNatKind(n__0) -> tt 316.83/292.17 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 316.83/292.17 isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 316.83/292.17 plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) 316.83/292.17 plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) 316.83/292.17 x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 316.83/292.17 0 -> n__0 316.83/292.17 plus(X1, X2) -> n__plus(X1, X2) 316.83/292.17 isNatKind(X) -> n__isNatKind(X) 316.83/292.17 s(X) -> n__s(X) 316.83/292.17 x(X1, X2) -> n__x(X1, X2) 316.83/292.17 and(X1, X2) -> n__and(X1, X2) 316.83/292.17 activate(n__0) -> 0 316.83/292.17 activate(n__plus(X1, X2)) -> plus(X1, X2) 316.83/292.17 activate(n__isNatKind(X)) -> isNatKind(X) 316.83/292.17 activate(n__s(X)) -> s(X) 316.83/292.17 activate(n__x(X1, X2)) -> x(X1, X2) 316.83/292.17 activate(n__and(X1, X2)) -> and(X1, X2) 316.83/292.17 activate(X) -> X 316.83/292.17 316.83/292.17 S is empty. 316.83/292.17 Rewrite Strategy: FULL 316.91/292.20 EOF