3.87/1.84 WORST_CASE(NON_POLY, ?) 3.87/1.84 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.87/1.84 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.87/1.84 3.87/1.84 3.87/1.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.87/1.84 3.87/1.84 (0) CpxTRS 3.87/1.84 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 3.87/1.84 (2) TRS for Loop Detection 3.87/1.84 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 3.87/1.84 (4) BEST 3.87/1.84 (5) proven lower bound 3.87/1.84 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 3.87/1.84 (7) BOUNDS(n^1, INF) 3.87/1.84 (8) TRS for Loop Detection 3.87/1.84 (9) DecreasingLoopProof [FINISHED, 87 ms] 3.87/1.84 (10) BOUNDS(EXP, INF) 3.87/1.84 3.87/1.84 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (0) 3.87/1.84 Obligation: 3.87/1.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.87/1.84 3.87/1.84 3.87/1.84 The TRS R consists of the following rules: 3.87/1.84 3.87/1.84 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 3.87/1.84 U12(tt, V2) -> U13(isNat(activate(V2))) 3.87/1.84 U13(tt) -> tt 3.87/1.84 U21(tt, V1) -> U22(isNat(activate(V1))) 3.87/1.84 U22(tt) -> tt 3.87/1.84 U31(tt, N) -> activate(N) 3.87/1.84 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 3.87/1.84 and(tt, X) -> activate(X) 3.87/1.84 isNat(n__0) -> tt 3.87/1.84 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 3.87/1.84 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 3.87/1.84 isNatKind(n__0) -> tt 3.87/1.84 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 3.87/1.84 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 3.87/1.84 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 3.87/1.84 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 3.87/1.84 0 -> n__0 3.87/1.84 plus(X1, X2) -> n__plus(X1, X2) 3.87/1.84 isNatKind(X) -> n__isNatKind(X) 3.87/1.84 s(X) -> n__s(X) 3.87/1.84 and(X1, X2) -> n__and(X1, X2) 3.87/1.84 isNat(X) -> n__isNat(X) 3.87/1.84 activate(n__0) -> 0 3.87/1.84 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 3.87/1.84 activate(n__isNatKind(X)) -> isNatKind(X) 3.87/1.84 activate(n__s(X)) -> s(activate(X)) 3.87/1.84 activate(n__and(X1, X2)) -> and(activate(X1), X2) 3.87/1.84 activate(n__isNat(X)) -> isNat(X) 3.87/1.84 activate(X) -> X 3.87/1.84 3.87/1.84 S is empty. 3.87/1.84 Rewrite Strategy: FULL 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 3.87/1.84 Transformed a relative TRS into a decreasing-loop problem. 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (2) 3.87/1.84 Obligation: 3.87/1.84 Analyzing the following TRS for decreasing loops: 3.87/1.84 3.87/1.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.87/1.84 3.87/1.84 3.87/1.84 The TRS R consists of the following rules: 3.87/1.84 3.87/1.84 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 3.87/1.84 U12(tt, V2) -> U13(isNat(activate(V2))) 3.87/1.84 U13(tt) -> tt 3.87/1.84 U21(tt, V1) -> U22(isNat(activate(V1))) 3.87/1.84 U22(tt) -> tt 3.87/1.84 U31(tt, N) -> activate(N) 3.87/1.84 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 3.87/1.84 and(tt, X) -> activate(X) 3.87/1.84 isNat(n__0) -> tt 3.87/1.84 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 3.87/1.84 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 3.87/1.84 isNatKind(n__0) -> tt 3.87/1.84 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 3.87/1.84 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 3.87/1.84 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 3.87/1.84 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 3.87/1.84 0 -> n__0 3.87/1.84 plus(X1, X2) -> n__plus(X1, X2) 3.87/1.84 isNatKind(X) -> n__isNatKind(X) 3.87/1.84 s(X) -> n__s(X) 3.87/1.84 and(X1, X2) -> n__and(X1, X2) 3.87/1.84 isNat(X) -> n__isNat(X) 3.87/1.84 activate(n__0) -> 0 3.87/1.84 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 3.87/1.84 activate(n__isNatKind(X)) -> isNatKind(X) 3.87/1.84 activate(n__s(X)) -> s(activate(X)) 3.87/1.84 activate(n__and(X1, X2)) -> and(activate(X1), X2) 3.87/1.84 activate(n__isNat(X)) -> isNat(X) 3.87/1.84 activate(X) -> X 3.87/1.84 3.87/1.84 S is empty. 3.87/1.84 Rewrite Strategy: FULL 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (3) DecreasingLoopProof (LOWER BOUND(ID)) 3.87/1.84 The following loop(s) give(s) rise to the lower bound Omega(n^1): 3.87/1.84 3.87/1.84 The rewrite sequence 3.87/1.84 3.87/1.84 activate(n__s(X)) ->^+ s(activate(X)) 3.87/1.84 3.87/1.84 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 3.87/1.84 3.87/1.84 The pumping substitution is [X / n__s(X)]. 3.87/1.84 3.87/1.84 The result substitution is [ ]. 3.87/1.84 3.87/1.84 3.87/1.84 3.87/1.84 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (4) 3.87/1.84 Complex Obligation (BEST) 3.87/1.84 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (5) 3.87/1.84 Obligation: 3.87/1.84 Proved the lower bound n^1 for the following obligation: 3.87/1.84 3.87/1.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.87/1.84 3.87/1.84 3.87/1.84 The TRS R consists of the following rules: 3.87/1.84 3.87/1.84 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 3.87/1.84 U12(tt, V2) -> U13(isNat(activate(V2))) 3.87/1.84 U13(tt) -> tt 3.87/1.84 U21(tt, V1) -> U22(isNat(activate(V1))) 3.87/1.84 U22(tt) -> tt 3.87/1.84 U31(tt, N) -> activate(N) 3.87/1.84 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 3.87/1.84 and(tt, X) -> activate(X) 3.87/1.84 isNat(n__0) -> tt 3.87/1.84 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 3.87/1.84 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 3.87/1.84 isNatKind(n__0) -> tt 3.87/1.84 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 3.87/1.84 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 3.87/1.84 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 3.87/1.84 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 3.87/1.84 0 -> n__0 3.87/1.84 plus(X1, X2) -> n__plus(X1, X2) 3.87/1.84 isNatKind(X) -> n__isNatKind(X) 3.87/1.84 s(X) -> n__s(X) 3.87/1.84 and(X1, X2) -> n__and(X1, X2) 3.87/1.84 isNat(X) -> n__isNat(X) 3.87/1.84 activate(n__0) -> 0 3.87/1.84 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 3.87/1.84 activate(n__isNatKind(X)) -> isNatKind(X) 3.87/1.84 activate(n__s(X)) -> s(activate(X)) 3.87/1.84 activate(n__and(X1, X2)) -> and(activate(X1), X2) 3.87/1.84 activate(n__isNat(X)) -> isNat(X) 3.87/1.84 activate(X) -> X 3.87/1.84 3.87/1.84 S is empty. 3.87/1.84 Rewrite Strategy: FULL 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (6) LowerBoundPropagationProof (FINISHED) 3.87/1.84 Propagated lower bound. 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (7) 3.87/1.84 BOUNDS(n^1, INF) 3.87/1.84 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (8) 3.87/1.84 Obligation: 3.87/1.84 Analyzing the following TRS for decreasing loops: 3.87/1.84 3.87/1.84 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.87/1.84 3.87/1.84 3.87/1.84 The TRS R consists of the following rules: 3.87/1.84 3.87/1.84 U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) 3.87/1.84 U12(tt, V2) -> U13(isNat(activate(V2))) 3.87/1.84 U13(tt) -> tt 3.87/1.84 U21(tt, V1) -> U22(isNat(activate(V1))) 3.87/1.84 U22(tt) -> tt 3.87/1.84 U31(tt, N) -> activate(N) 3.87/1.84 U41(tt, M, N) -> s(plus(activate(N), activate(M))) 3.87/1.84 and(tt, X) -> activate(X) 3.87/1.84 isNat(n__0) -> tt 3.87/1.84 isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) 3.87/1.84 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 3.87/1.84 isNatKind(n__0) -> tt 3.87/1.84 isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) 3.87/1.84 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 3.87/1.84 plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) 3.87/1.84 plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 3.87/1.84 0 -> n__0 3.87/1.84 plus(X1, X2) -> n__plus(X1, X2) 3.87/1.84 isNatKind(X) -> n__isNatKind(X) 3.87/1.84 s(X) -> n__s(X) 3.87/1.84 and(X1, X2) -> n__and(X1, X2) 3.87/1.84 isNat(X) -> n__isNat(X) 3.87/1.84 activate(n__0) -> 0 3.87/1.84 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 3.87/1.84 activate(n__isNatKind(X)) -> isNatKind(X) 3.87/1.84 activate(n__s(X)) -> s(activate(X)) 3.87/1.84 activate(n__and(X1, X2)) -> and(activate(X1), X2) 3.87/1.84 activate(n__isNat(X)) -> isNat(X) 3.87/1.84 activate(X) -> X 3.87/1.84 3.87/1.84 S is empty. 3.87/1.84 Rewrite Strategy: FULL 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (9) DecreasingLoopProof (FINISHED) 3.87/1.84 The following loop(s) give(s) rise to the lower bound EXP: 3.87/1.84 3.87/1.84 The rewrite sequence 3.87/1.84 3.87/1.84 activate(n__isNat(n__s(V11_0))) ->^+ U21(isNatKind(activate(V11_0)), activate(V11_0)) 3.87/1.84 3.87/1.84 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 3.87/1.84 3.87/1.84 The pumping substitution is [V11_0 / n__isNat(n__s(V11_0))]. 3.87/1.84 3.87/1.84 The result substitution is [ ]. 3.87/1.84 3.87/1.84 3.87/1.84 3.87/1.84 The rewrite sequence 3.87/1.84 3.87/1.84 activate(n__isNat(n__s(V11_0))) ->^+ U21(isNatKind(activate(V11_0)), activate(V11_0)) 3.87/1.84 3.87/1.84 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 3.87/1.84 3.87/1.84 The pumping substitution is [V11_0 / n__isNat(n__s(V11_0))]. 3.87/1.84 3.87/1.84 The result substitution is [ ]. 3.87/1.84 3.87/1.84 3.87/1.84 3.87/1.84 3.87/1.84 ---------------------------------------- 3.87/1.84 3.87/1.84 (10) 3.87/1.84 BOUNDS(EXP, INF) 3.87/1.87 EOF