317.24/291.54 WORST_CASE(Omega(n^1), ?) 317.24/291.55 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 317.24/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 317.24/291.55 317.24/291.55 317.24/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.24/291.55 317.24/291.55 (0) CpxTRS 317.24/291.55 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 317.24/291.55 (2) TRS for Loop Detection 317.24/291.55 (3) DecreasingLoopProof [LOWER BOUND(ID), 195 ms] 317.24/291.55 (4) BEST 317.24/291.55 (5) proven lower bound 317.24/291.55 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 317.24/291.55 (7) BOUNDS(n^1, INF) 317.24/291.55 (8) TRS for Loop Detection 317.24/291.55 317.24/291.55 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (0) 317.24/291.55 Obligation: 317.24/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.24/291.55 317.24/291.55 317.24/291.55 The TRS R consists of the following rules: 317.24/291.55 317.24/291.55 U101(tt, M, N) -> U102(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U102(tt, M, N) -> U103(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U103(tt, M, N) -> U104(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U104(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 317.24/291.55 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 317.24/291.55 U15(tt, V2) -> U16(isNat(activate(V2))) 317.24/291.55 U16(tt) -> tt 317.24/291.55 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 U22(tt, V1) -> U23(isNat(activate(V1))) 317.24/291.55 U23(tt) -> tt 317.24/291.55 U31(tt, V1, V2) -> U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U32(tt, V1, V2) -> U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U33(tt, V1, V2) -> U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U34(tt, V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 317.24/291.55 U35(tt, V2) -> U36(isNat(activate(V2))) 317.24/291.55 U36(tt) -> tt 317.24/291.55 U41(tt, V2) -> U42(isNatKind(activate(V2))) 317.24/291.55 U42(tt) -> tt 317.24/291.55 U51(tt) -> tt 317.24/291.55 U61(tt, V2) -> U62(isNatKind(activate(V2))) 317.24/291.55 U62(tt) -> tt 317.24/291.55 U71(tt, N) -> U72(isNatKind(activate(N)), activate(N)) 317.24/291.55 U72(tt, N) -> activate(N) 317.24/291.55 U81(tt, M, N) -> U82(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U82(tt, M, N) -> U83(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U83(tt, M, N) -> U84(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U84(tt, M, N) -> s(plus(activate(N), activate(M))) 317.24/291.55 U91(tt, N) -> U92(isNatKind(activate(N))) 317.24/291.55 U92(tt) -> 0 317.24/291.55 isNat(n__0) -> tt 317.24/291.55 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 isNat(n__x(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNatKind(n__0) -> tt 317.24/291.55 isNatKind(n__plus(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 317.24/291.55 isNatKind(n__x(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 plus(N, 0) -> U71(isNat(N), N) 317.24/291.55 plus(N, s(M)) -> U81(isNat(M), M, N) 317.24/291.55 x(N, 0) -> U91(isNat(N), N) 317.24/291.55 x(N, s(M)) -> U101(isNat(M), M, N) 317.24/291.55 0 -> n__0 317.24/291.55 plus(X1, X2) -> n__plus(X1, X2) 317.24/291.55 s(X) -> n__s(X) 317.24/291.55 x(X1, X2) -> n__x(X1, X2) 317.24/291.55 activate(n__0) -> 0 317.24/291.55 activate(n__plus(X1, X2)) -> plus(X1, X2) 317.24/291.55 activate(n__s(X)) -> s(X) 317.24/291.55 activate(n__x(X1, X2)) -> x(X1, X2) 317.24/291.55 activate(X) -> X 317.24/291.55 317.24/291.55 S is empty. 317.24/291.55 Rewrite Strategy: FULL 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 317.24/291.55 Transformed a relative TRS into a decreasing-loop problem. 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (2) 317.24/291.55 Obligation: 317.24/291.55 Analyzing the following TRS for decreasing loops: 317.24/291.55 317.24/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.24/291.55 317.24/291.55 317.24/291.55 The TRS R consists of the following rules: 317.24/291.55 317.24/291.55 U101(tt, M, N) -> U102(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U102(tt, M, N) -> U103(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U103(tt, M, N) -> U104(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U104(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 317.24/291.55 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 317.24/291.55 U15(tt, V2) -> U16(isNat(activate(V2))) 317.24/291.55 U16(tt) -> tt 317.24/291.55 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 U22(tt, V1) -> U23(isNat(activate(V1))) 317.24/291.55 U23(tt) -> tt 317.24/291.55 U31(tt, V1, V2) -> U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U32(tt, V1, V2) -> U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U33(tt, V1, V2) -> U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U34(tt, V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 317.24/291.55 U35(tt, V2) -> U36(isNat(activate(V2))) 317.24/291.55 U36(tt) -> tt 317.24/291.55 U41(tt, V2) -> U42(isNatKind(activate(V2))) 317.24/291.55 U42(tt) -> tt 317.24/291.55 U51(tt) -> tt 317.24/291.55 U61(tt, V2) -> U62(isNatKind(activate(V2))) 317.24/291.55 U62(tt) -> tt 317.24/291.55 U71(tt, N) -> U72(isNatKind(activate(N)), activate(N)) 317.24/291.55 U72(tt, N) -> activate(N) 317.24/291.55 U81(tt, M, N) -> U82(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U82(tt, M, N) -> U83(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U83(tt, M, N) -> U84(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U84(tt, M, N) -> s(plus(activate(N), activate(M))) 317.24/291.55 U91(tt, N) -> U92(isNatKind(activate(N))) 317.24/291.55 U92(tt) -> 0 317.24/291.55 isNat(n__0) -> tt 317.24/291.55 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 isNat(n__x(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNatKind(n__0) -> tt 317.24/291.55 isNatKind(n__plus(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 317.24/291.55 isNatKind(n__x(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 plus(N, 0) -> U71(isNat(N), N) 317.24/291.55 plus(N, s(M)) -> U81(isNat(M), M, N) 317.24/291.55 x(N, 0) -> U91(isNat(N), N) 317.24/291.55 x(N, s(M)) -> U101(isNat(M), M, N) 317.24/291.55 0 -> n__0 317.24/291.55 plus(X1, X2) -> n__plus(X1, X2) 317.24/291.55 s(X) -> n__s(X) 317.24/291.55 x(X1, X2) -> n__x(X1, X2) 317.24/291.55 activate(n__0) -> 0 317.24/291.55 activate(n__plus(X1, X2)) -> plus(X1, X2) 317.24/291.55 activate(n__s(X)) -> s(X) 317.24/291.55 activate(n__x(X1, X2)) -> x(X1, X2) 317.24/291.55 activate(X) -> X 317.24/291.55 317.24/291.55 S is empty. 317.24/291.55 Rewrite Strategy: FULL 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (3) DecreasingLoopProof (LOWER BOUND(ID)) 317.24/291.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 317.24/291.55 317.24/291.55 The rewrite sequence 317.24/291.55 317.24/291.55 isNatKind(n__x(V1, V2)) ->^+ U61(isNatKind(V1), activate(V2)) 317.24/291.55 317.24/291.55 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 317.24/291.55 317.24/291.55 The pumping substitution is [V1 / n__x(V1, V2)]. 317.24/291.55 317.24/291.55 The result substitution is [ ]. 317.24/291.55 317.24/291.55 317.24/291.55 317.24/291.55 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (4) 317.24/291.55 Complex Obligation (BEST) 317.24/291.55 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (5) 317.24/291.55 Obligation: 317.24/291.55 Proved the lower bound n^1 for the following obligation: 317.24/291.55 317.24/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.24/291.55 317.24/291.55 317.24/291.55 The TRS R consists of the following rules: 317.24/291.55 317.24/291.55 U101(tt, M, N) -> U102(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U102(tt, M, N) -> U103(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U103(tt, M, N) -> U104(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U104(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 317.24/291.55 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 317.24/291.55 U15(tt, V2) -> U16(isNat(activate(V2))) 317.24/291.55 U16(tt) -> tt 317.24/291.55 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 U22(tt, V1) -> U23(isNat(activate(V1))) 317.24/291.55 U23(tt) -> tt 317.24/291.55 U31(tt, V1, V2) -> U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U32(tt, V1, V2) -> U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U33(tt, V1, V2) -> U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U34(tt, V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 317.24/291.55 U35(tt, V2) -> U36(isNat(activate(V2))) 317.24/291.55 U36(tt) -> tt 317.24/291.55 U41(tt, V2) -> U42(isNatKind(activate(V2))) 317.24/291.55 U42(tt) -> tt 317.24/291.55 U51(tt) -> tt 317.24/291.55 U61(tt, V2) -> U62(isNatKind(activate(V2))) 317.24/291.55 U62(tt) -> tt 317.24/291.55 U71(tt, N) -> U72(isNatKind(activate(N)), activate(N)) 317.24/291.55 U72(tt, N) -> activate(N) 317.24/291.55 U81(tt, M, N) -> U82(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U82(tt, M, N) -> U83(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U83(tt, M, N) -> U84(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U84(tt, M, N) -> s(plus(activate(N), activate(M))) 317.24/291.55 U91(tt, N) -> U92(isNatKind(activate(N))) 317.24/291.55 U92(tt) -> 0 317.24/291.55 isNat(n__0) -> tt 317.24/291.55 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 isNat(n__x(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNatKind(n__0) -> tt 317.24/291.55 isNatKind(n__plus(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 317.24/291.55 isNatKind(n__x(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 plus(N, 0) -> U71(isNat(N), N) 317.24/291.55 plus(N, s(M)) -> U81(isNat(M), M, N) 317.24/291.55 x(N, 0) -> U91(isNat(N), N) 317.24/291.55 x(N, s(M)) -> U101(isNat(M), M, N) 317.24/291.55 0 -> n__0 317.24/291.55 plus(X1, X2) -> n__plus(X1, X2) 317.24/291.55 s(X) -> n__s(X) 317.24/291.55 x(X1, X2) -> n__x(X1, X2) 317.24/291.55 activate(n__0) -> 0 317.24/291.55 activate(n__plus(X1, X2)) -> plus(X1, X2) 317.24/291.55 activate(n__s(X)) -> s(X) 317.24/291.55 activate(n__x(X1, X2)) -> x(X1, X2) 317.24/291.55 activate(X) -> X 317.24/291.55 317.24/291.55 S is empty. 317.24/291.55 Rewrite Strategy: FULL 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (6) LowerBoundPropagationProof (FINISHED) 317.24/291.55 Propagated lower bound. 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (7) 317.24/291.55 BOUNDS(n^1, INF) 317.24/291.55 317.24/291.55 ---------------------------------------- 317.24/291.55 317.24/291.55 (8) 317.24/291.55 Obligation: 317.24/291.55 Analyzing the following TRS for decreasing loops: 317.24/291.55 317.24/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.24/291.55 317.24/291.55 317.24/291.55 The TRS R consists of the following rules: 317.24/291.55 317.24/291.55 U101(tt, M, N) -> U102(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U102(tt, M, N) -> U103(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U103(tt, M, N) -> U104(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U104(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 317.24/291.55 U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) 317.24/291.55 U15(tt, V2) -> U16(isNat(activate(V2))) 317.24/291.55 U16(tt) -> tt 317.24/291.55 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 U22(tt, V1) -> U23(isNat(activate(V1))) 317.24/291.55 U23(tt) -> tt 317.24/291.55 U31(tt, V1, V2) -> U32(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 U32(tt, V1, V2) -> U33(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U33(tt, V1, V2) -> U34(isNatKind(activate(V2)), activate(V1), activate(V2)) 317.24/291.55 U34(tt, V1, V2) -> U35(isNat(activate(V1)), activate(V2)) 317.24/291.55 U35(tt, V2) -> U36(isNat(activate(V2))) 317.24/291.55 U36(tt) -> tt 317.24/291.55 U41(tt, V2) -> U42(isNatKind(activate(V2))) 317.24/291.55 U42(tt) -> tt 317.24/291.55 U51(tt) -> tt 317.24/291.55 U61(tt, V2) -> U62(isNatKind(activate(V2))) 317.24/291.55 U62(tt) -> tt 317.24/291.55 U71(tt, N) -> U72(isNatKind(activate(N)), activate(N)) 317.24/291.55 U72(tt, N) -> activate(N) 317.24/291.55 U81(tt, M, N) -> U82(isNatKind(activate(M)), activate(M), activate(N)) 317.24/291.55 U82(tt, M, N) -> U83(isNat(activate(N)), activate(M), activate(N)) 317.24/291.55 U83(tt, M, N) -> U84(isNatKind(activate(N)), activate(M), activate(N)) 317.24/291.55 U84(tt, M, N) -> s(plus(activate(N), activate(M))) 317.24/291.55 U91(tt, N) -> U92(isNatKind(activate(N))) 317.24/291.55 U92(tt) -> 0 317.24/291.55 isNat(n__0) -> tt 317.24/291.55 isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.24/291.55 isNat(n__x(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V1), activate(V2)) 317.24/291.55 isNatKind(n__0) -> tt 317.24/291.55 isNatKind(n__plus(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) 317.24/291.55 isNatKind(n__x(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 317.24/291.55 plus(N, 0) -> U71(isNat(N), N) 317.24/291.55 plus(N, s(M)) -> U81(isNat(M), M, N) 317.24/291.55 x(N, 0) -> U91(isNat(N), N) 317.24/291.55 x(N, s(M)) -> U101(isNat(M), M, N) 317.24/291.55 0 -> n__0 317.24/291.55 plus(X1, X2) -> n__plus(X1, X2) 317.24/291.55 s(X) -> n__s(X) 317.24/291.55 x(X1, X2) -> n__x(X1, X2) 317.24/291.55 activate(n__0) -> 0 317.24/291.55 activate(n__plus(X1, X2)) -> plus(X1, X2) 317.24/291.55 activate(n__s(X)) -> s(X) 317.24/291.55 activate(n__x(X1, X2)) -> x(X1, X2) 317.24/291.55 activate(X) -> X 317.24/291.55 317.24/291.55 S is empty. 317.24/291.55 Rewrite Strategy: FULL 317.24/291.58 EOF