6.43/2.37 WORST_CASE(NON_POLY, ?) 6.43/2.45 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 6.43/2.45 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.43/2.45 6.43/2.45 6.43/2.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 6.43/2.45 6.43/2.45 (0) CpxTRS 6.43/2.45 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 6.43/2.45 (2) TRS for Loop Detection 6.43/2.45 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 6.43/2.45 (4) BEST 6.43/2.45 (5) proven lower bound 6.43/2.45 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 6.43/2.45 (7) BOUNDS(n^1, INF) 6.43/2.45 (8) TRS for Loop Detection 6.43/2.45 (9) DecreasingLoopProof [FINISHED, 532 ms] 6.43/2.45 (10) BOUNDS(EXP, INF) 6.43/2.45 6.43/2.45 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (0) 6.43/2.45 Obligation: 6.43/2.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 6.43/2.45 6.43/2.45 6.43/2.45 The TRS R consists of the following rules: 6.43/2.45 6.43/2.45 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 6.43/2.45 U15(tt, V2) -> U16(isNat(activate(V2))) 6.43/2.45 U16(tt) -> tt 6.43/2.45 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 U22(tt, V1) -> U23(isNat(activate(V1))) 6.43/2.45 U23(tt) -> tt 6.43/2.45 U31(tt, V2) -> U32(isNatKind(activate(V2))) 6.43/2.45 U32(tt) -> tt 6.43/2.45 U41(tt) -> tt 6.43/2.45 U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) 6.43/2.45 U52(tt, N) -> activate(N) 6.43/2.45 U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) 6.43/2.45 U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) 6.43/2.45 U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) 6.43/2.45 U64(tt, M, N) -> s(plus(activate(N), activate(M))) 6.43/2.45 isNat(n__0) -> tt 6.43/2.45 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 isNatKind(n__0) -> tt 6.43/2.45 isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) 6.43/2.45 isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 6.43/2.45 plus(N, 0) -> U51(isNat(N), N) 6.43/2.45 plus(N, s(M)) -> U61(isNat(M), M, N) 6.43/2.45 0 -> n__0 6.43/2.45 plus(X1, X2) -> n__plus(X1, X2) 6.43/2.45 s(X) -> n__s(X) 6.43/2.45 activate(n__0) -> 0 6.43/2.45 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 6.43/2.45 activate(n__s(X)) -> s(activate(X)) 6.43/2.45 activate(X) -> X 6.43/2.45 6.43/2.45 S is empty. 6.43/2.45 Rewrite Strategy: FULL 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 6.43/2.45 Transformed a relative TRS into a decreasing-loop problem. 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (2) 6.43/2.45 Obligation: 6.43/2.45 Analyzing the following TRS for decreasing loops: 6.43/2.45 6.43/2.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 6.43/2.45 6.43/2.45 6.43/2.45 The TRS R consists of the following rules: 6.43/2.45 6.43/2.45 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 6.43/2.45 U15(tt, V2) -> U16(isNat(activate(V2))) 6.43/2.45 U16(tt) -> tt 6.43/2.45 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 U22(tt, V1) -> U23(isNat(activate(V1))) 6.43/2.45 U23(tt) -> tt 6.43/2.45 U31(tt, V2) -> U32(isNatKind(activate(V2))) 6.43/2.45 U32(tt) -> tt 6.43/2.45 U41(tt) -> tt 6.43/2.45 U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) 6.43/2.45 U52(tt, N) -> activate(N) 6.43/2.45 U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) 6.43/2.45 U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) 6.43/2.45 U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) 6.43/2.45 U64(tt, M, N) -> s(plus(activate(N), activate(M))) 6.43/2.45 isNat(n__0) -> tt 6.43/2.45 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 isNatKind(n__0) -> tt 6.43/2.45 isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) 6.43/2.45 isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 6.43/2.45 plus(N, 0) -> U51(isNat(N), N) 6.43/2.45 plus(N, s(M)) -> U61(isNat(M), M, N) 6.43/2.45 0 -> n__0 6.43/2.45 plus(X1, X2) -> n__plus(X1, X2) 6.43/2.45 s(X) -> n__s(X) 6.43/2.45 activate(n__0) -> 0 6.43/2.45 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 6.43/2.45 activate(n__s(X)) -> s(activate(X)) 6.43/2.45 activate(X) -> X 6.43/2.45 6.43/2.45 S is empty. 6.43/2.45 Rewrite Strategy: FULL 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (3) DecreasingLoopProof (LOWER BOUND(ID)) 6.43/2.45 The following loop(s) give(s) rise to the lower bound Omega(n^1): 6.43/2.45 6.43/2.45 The rewrite sequence 6.43/2.45 6.43/2.45 activate(n__s(X)) ->^+ s(activate(X)) 6.43/2.45 6.43/2.45 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 6.43/2.45 6.43/2.45 The pumping substitution is [X / n__s(X)]. 6.43/2.45 6.43/2.45 The result substitution is [ ]. 6.43/2.45 6.43/2.45 6.43/2.45 6.43/2.45 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (4) 6.43/2.45 Complex Obligation (BEST) 6.43/2.45 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (5) 6.43/2.45 Obligation: 6.43/2.45 Proved the lower bound n^1 for the following obligation: 6.43/2.45 6.43/2.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 6.43/2.45 6.43/2.45 6.43/2.45 The TRS R consists of the following rules: 6.43/2.45 6.43/2.45 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 6.43/2.45 U15(tt, V2) -> U16(isNat(activate(V2))) 6.43/2.45 U16(tt) -> tt 6.43/2.45 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 U22(tt, V1) -> U23(isNat(activate(V1))) 6.43/2.45 U23(tt) -> tt 6.43/2.45 U31(tt, V2) -> U32(isNatKind(activate(V2))) 6.43/2.45 U32(tt) -> tt 6.43/2.45 U41(tt) -> tt 6.43/2.45 U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) 6.43/2.45 U52(tt, N) -> activate(N) 6.43/2.45 U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) 6.43/2.45 U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) 6.43/2.45 U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) 6.43/2.45 U64(tt, M, N) -> s(plus(activate(N), activate(M))) 6.43/2.45 isNat(n__0) -> tt 6.43/2.45 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 isNatKind(n__0) -> tt 6.43/2.45 isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) 6.43/2.45 isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 6.43/2.45 plus(N, 0) -> U51(isNat(N), N) 6.43/2.45 plus(N, s(M)) -> U61(isNat(M), M, N) 6.43/2.45 0 -> n__0 6.43/2.45 plus(X1, X2) -> n__plus(X1, X2) 6.43/2.45 s(X) -> n__s(X) 6.43/2.45 activate(n__0) -> 0 6.43/2.45 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 6.43/2.45 activate(n__s(X)) -> s(activate(X)) 6.43/2.45 activate(X) -> X 6.43/2.45 6.43/2.45 S is empty. 6.43/2.45 Rewrite Strategy: FULL 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (6) LowerBoundPropagationProof (FINISHED) 6.43/2.45 Propagated lower bound. 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (7) 6.43/2.45 BOUNDS(n^1, INF) 6.43/2.45 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (8) 6.43/2.45 Obligation: 6.43/2.45 Analyzing the following TRS for decreasing loops: 6.43/2.45 6.43/2.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 6.43/2.45 6.43/2.45 6.43/2.45 The TRS R consists of the following rules: 6.43/2.45 6.43/2.45 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 6.43/2.45 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 6.43/2.45 U15(tt, V2) -> U16(isNat(activate(V2))) 6.43/2.45 U16(tt) -> tt 6.43/2.45 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 U22(tt, V1) -> U23(isNat(activate(V1))) 6.43/2.45 U23(tt) -> tt 6.43/2.45 U31(tt, V2) -> U32(isNatKind(activate(V2))) 6.43/2.45 U32(tt) -> tt 6.43/2.45 U41(tt) -> tt 6.43/2.45 U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) 6.43/2.45 U52(tt, N) -> activate(N) 6.43/2.45 U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) 6.43/2.45 U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) 6.43/2.45 U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) 6.43/2.45 U64(tt, M, N) -> s(plus(activate(N), activate(M))) 6.43/2.45 isNat(n__0) -> tt 6.43/2.45 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 6.43/2.45 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 6.43/2.45 isNatKind(n__0) -> tt 6.43/2.45 isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) 6.43/2.45 isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) 6.43/2.45 plus(N, 0) -> U51(isNat(N), N) 6.43/2.45 plus(N, s(M)) -> U61(isNat(M), M, N) 6.43/2.45 0 -> n__0 6.43/2.45 plus(X1, X2) -> n__plus(X1, X2) 6.43/2.45 s(X) -> n__s(X) 6.43/2.45 activate(n__0) -> 0 6.43/2.45 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) 6.43/2.45 activate(n__s(X)) -> s(activate(X)) 6.43/2.45 activate(X) -> X 6.43/2.45 6.43/2.45 S is empty. 6.43/2.45 Rewrite Strategy: FULL 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (9) DecreasingLoopProof (FINISHED) 6.43/2.45 The following loop(s) give(s) rise to the lower bound EXP: 6.43/2.45 6.43/2.45 The rewrite sequence 6.43/2.45 6.43/2.45 activate(n__plus(X1, n__s(X1_0))) ->^+ U61(isNat(activate(X1_0)), activate(X1_0), activate(X1)) 6.43/2.45 6.43/2.45 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 6.43/2.45 6.43/2.45 The pumping substitution is [X1_0 / n__plus(X1, n__s(X1_0))]. 6.43/2.45 6.43/2.45 The result substitution is [ ]. 6.43/2.45 6.43/2.45 6.43/2.45 6.43/2.45 The rewrite sequence 6.43/2.45 6.43/2.45 activate(n__plus(X1, n__s(X1_0))) ->^+ U61(isNat(activate(X1_0)), activate(X1_0), activate(X1)) 6.43/2.45 6.43/2.45 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 6.43/2.45 6.43/2.45 The pumping substitution is [X1_0 / n__plus(X1, n__s(X1_0))]. 6.43/2.45 6.43/2.45 The result substitution is [ ]. 6.43/2.45 6.43/2.45 6.43/2.45 6.43/2.45 6.43/2.45 ---------------------------------------- 6.43/2.45 6.43/2.45 (10) 6.43/2.45 BOUNDS(EXP, INF) 6.73/2.47 EOF