8.30/2.89 WORST_CASE(NON_POLY, ?) 8.51/2.97 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 8.51/2.97 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 8.51/2.97 8.51/2.97 8.51/2.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.51/2.97 8.51/2.97 (0) CpxTRS 8.51/2.97 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 8.51/2.97 (2) TRS for Loop Detection 8.51/2.97 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 8.51/2.97 (4) BEST 8.51/2.97 (5) proven lower bound 8.51/2.97 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 8.51/2.97 (7) BOUNDS(n^1, INF) 8.51/2.97 (8) TRS for Loop Detection 8.51/2.97 (9) DecreasingLoopProof [FINISHED, 899 ms] 8.51/2.97 (10) BOUNDS(EXP, INF) 8.51/2.97 8.51/2.97 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (0) 8.51/2.97 Obligation: 8.51/2.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.51/2.97 8.51/2.97 8.51/2.97 The TRS R consists of the following rules: 8.51/2.97 8.51/2.97 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 8.51/2.97 a__U12(tt) -> tt 8.51/2.97 a__U21(tt) -> tt 8.51/2.97 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 8.51/2.97 a__U32(tt) -> tt 8.51/2.97 a__U41(tt, N) -> mark(N) 8.51/2.97 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 8.51/2.97 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 8.51/2.97 a__U61(tt) -> 0 8.51/2.97 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 8.51/2.97 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 8.51/2.97 a__isNat(0) -> tt 8.51/2.97 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 8.51/2.97 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 8.51/2.97 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 8.51/2.97 a__plus(N, 0) -> a__U41(a__isNat(N), N) 8.51/2.97 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 8.51/2.97 a__x(N, 0) -> a__U61(a__isNat(N)) 8.51/2.97 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 8.51/2.97 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 8.51/2.97 mark(U12(X)) -> a__U12(mark(X)) 8.51/2.97 mark(isNat(X)) -> a__isNat(X) 8.51/2.97 mark(U21(X)) -> a__U21(mark(X)) 8.51/2.97 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 8.51/2.97 mark(U32(X)) -> a__U32(mark(X)) 8.51/2.97 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 8.51/2.97 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 8.51/2.97 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 8.51/2.97 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 8.51/2.97 mark(U61(X)) -> a__U61(mark(X)) 8.51/2.97 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 8.51/2.97 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 8.51/2.97 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 8.51/2.97 mark(tt) -> tt 8.51/2.97 mark(s(X)) -> s(mark(X)) 8.51/2.97 mark(0) -> 0 8.51/2.97 a__U11(X1, X2) -> U11(X1, X2) 8.51/2.97 a__U12(X) -> U12(X) 8.51/2.97 a__isNat(X) -> isNat(X) 8.51/2.97 a__U21(X) -> U21(X) 8.51/2.97 a__U31(X1, X2) -> U31(X1, X2) 8.51/2.97 a__U32(X) -> U32(X) 8.51/2.97 a__U41(X1, X2) -> U41(X1, X2) 8.51/2.97 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 8.51/2.97 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 8.51/2.97 a__plus(X1, X2) -> plus(X1, X2) 8.51/2.97 a__U61(X) -> U61(X) 8.51/2.97 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 8.51/2.97 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 8.51/2.97 a__x(X1, X2) -> x(X1, X2) 8.51/2.97 8.51/2.97 S is empty. 8.51/2.97 Rewrite Strategy: INNERMOST 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 8.51/2.97 Transformed a relative TRS into a decreasing-loop problem. 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (2) 8.51/2.97 Obligation: 8.51/2.97 Analyzing the following TRS for decreasing loops: 8.51/2.97 8.51/2.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.51/2.97 8.51/2.97 8.51/2.97 The TRS R consists of the following rules: 8.51/2.97 8.51/2.97 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 8.51/2.97 a__U12(tt) -> tt 8.51/2.97 a__U21(tt) -> tt 8.51/2.97 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 8.51/2.97 a__U32(tt) -> tt 8.51/2.97 a__U41(tt, N) -> mark(N) 8.51/2.97 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 8.51/2.97 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 8.51/2.97 a__U61(tt) -> 0 8.51/2.97 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 8.51/2.97 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 8.51/2.97 a__isNat(0) -> tt 8.51/2.97 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 8.51/2.97 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 8.51/2.97 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 8.51/2.97 a__plus(N, 0) -> a__U41(a__isNat(N), N) 8.51/2.97 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 8.51/2.97 a__x(N, 0) -> a__U61(a__isNat(N)) 8.51/2.97 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 8.51/2.97 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 8.51/2.97 mark(U12(X)) -> a__U12(mark(X)) 8.51/2.97 mark(isNat(X)) -> a__isNat(X) 8.51/2.97 mark(U21(X)) -> a__U21(mark(X)) 8.51/2.97 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 8.51/2.97 mark(U32(X)) -> a__U32(mark(X)) 8.51/2.97 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 8.51/2.97 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 8.51/2.97 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 8.51/2.97 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 8.51/2.97 mark(U61(X)) -> a__U61(mark(X)) 8.51/2.97 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 8.51/2.97 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 8.51/2.97 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 8.51/2.97 mark(tt) -> tt 8.51/2.97 mark(s(X)) -> s(mark(X)) 8.51/2.97 mark(0) -> 0 8.51/2.97 a__U11(X1, X2) -> U11(X1, X2) 8.51/2.97 a__U12(X) -> U12(X) 8.51/2.97 a__isNat(X) -> isNat(X) 8.51/2.97 a__U21(X) -> U21(X) 8.51/2.97 a__U31(X1, X2) -> U31(X1, X2) 8.51/2.97 a__U32(X) -> U32(X) 8.51/2.97 a__U41(X1, X2) -> U41(X1, X2) 8.51/2.97 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 8.51/2.97 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 8.51/2.97 a__plus(X1, X2) -> plus(X1, X2) 8.51/2.97 a__U61(X) -> U61(X) 8.51/2.97 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 8.51/2.97 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 8.51/2.97 a__x(X1, X2) -> x(X1, X2) 8.51/2.97 8.51/2.97 S is empty. 8.51/2.97 Rewrite Strategy: INNERMOST 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (3) DecreasingLoopProof (LOWER BOUND(ID)) 8.51/2.97 The following loop(s) give(s) rise to the lower bound Omega(n^1): 8.51/2.97 8.51/2.97 The rewrite sequence 8.51/2.97 8.51/2.97 mark(U12(X)) ->^+ a__U12(mark(X)) 8.51/2.97 8.51/2.97 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 8.51/2.97 8.51/2.97 The pumping substitution is [X / U12(X)]. 8.51/2.97 8.51/2.97 The result substitution is [ ]. 8.51/2.97 8.51/2.97 8.51/2.97 8.51/2.97 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (4) 8.51/2.97 Complex Obligation (BEST) 8.51/2.97 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (5) 8.51/2.97 Obligation: 8.51/2.97 Proved the lower bound n^1 for the following obligation: 8.51/2.97 8.51/2.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.51/2.97 8.51/2.97 8.51/2.97 The TRS R consists of the following rules: 8.51/2.97 8.51/2.97 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 8.51/2.97 a__U12(tt) -> tt 8.51/2.97 a__U21(tt) -> tt 8.51/2.97 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 8.51/2.97 a__U32(tt) -> tt 8.51/2.97 a__U41(tt, N) -> mark(N) 8.51/2.97 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 8.51/2.97 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 8.51/2.97 a__U61(tt) -> 0 8.51/2.97 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 8.51/2.97 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 8.51/2.97 a__isNat(0) -> tt 8.51/2.97 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 8.51/2.97 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 8.51/2.97 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 8.51/2.97 a__plus(N, 0) -> a__U41(a__isNat(N), N) 8.51/2.97 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 8.51/2.97 a__x(N, 0) -> a__U61(a__isNat(N)) 8.51/2.97 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 8.51/2.97 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 8.51/2.97 mark(U12(X)) -> a__U12(mark(X)) 8.51/2.97 mark(isNat(X)) -> a__isNat(X) 8.51/2.97 mark(U21(X)) -> a__U21(mark(X)) 8.51/2.97 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 8.51/2.97 mark(U32(X)) -> a__U32(mark(X)) 8.51/2.97 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 8.51/2.97 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 8.51/2.97 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 8.51/2.97 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 8.51/2.97 mark(U61(X)) -> a__U61(mark(X)) 8.51/2.97 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 8.51/2.97 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 8.51/2.97 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 8.51/2.97 mark(tt) -> tt 8.51/2.97 mark(s(X)) -> s(mark(X)) 8.51/2.97 mark(0) -> 0 8.51/2.97 a__U11(X1, X2) -> U11(X1, X2) 8.51/2.97 a__U12(X) -> U12(X) 8.51/2.97 a__isNat(X) -> isNat(X) 8.51/2.97 a__U21(X) -> U21(X) 8.51/2.97 a__U31(X1, X2) -> U31(X1, X2) 8.51/2.97 a__U32(X) -> U32(X) 8.51/2.97 a__U41(X1, X2) -> U41(X1, X2) 8.51/2.97 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 8.51/2.97 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 8.51/2.97 a__plus(X1, X2) -> plus(X1, X2) 8.51/2.97 a__U61(X) -> U61(X) 8.51/2.97 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 8.51/2.97 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 8.51/2.97 a__x(X1, X2) -> x(X1, X2) 8.51/2.97 8.51/2.97 S is empty. 8.51/2.97 Rewrite Strategy: INNERMOST 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (6) LowerBoundPropagationProof (FINISHED) 8.51/2.97 Propagated lower bound. 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (7) 8.51/2.97 BOUNDS(n^1, INF) 8.51/2.97 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (8) 8.51/2.97 Obligation: 8.51/2.97 Analyzing the following TRS for decreasing loops: 8.51/2.97 8.51/2.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.51/2.97 8.51/2.97 8.51/2.97 The TRS R consists of the following rules: 8.51/2.97 8.51/2.97 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 8.51/2.97 a__U12(tt) -> tt 8.51/2.97 a__U21(tt) -> tt 8.51/2.97 a__U31(tt, V2) -> a__U32(a__isNat(V2)) 8.51/2.97 a__U32(tt) -> tt 8.51/2.97 a__U41(tt, N) -> mark(N) 8.51/2.97 a__U51(tt, M, N) -> a__U52(a__isNat(N), M, N) 8.51/2.97 a__U52(tt, M, N) -> s(a__plus(mark(N), mark(M))) 8.51/2.97 a__U61(tt) -> 0 8.51/2.97 a__U71(tt, M, N) -> a__U72(a__isNat(N), M, N) 8.51/2.97 a__U72(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 8.51/2.97 a__isNat(0) -> tt 8.51/2.97 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 8.51/2.97 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 8.51/2.97 a__isNat(x(V1, V2)) -> a__U31(a__isNat(V1), V2) 8.51/2.97 a__plus(N, 0) -> a__U41(a__isNat(N), N) 8.51/2.97 a__plus(N, s(M)) -> a__U51(a__isNat(M), M, N) 8.51/2.97 a__x(N, 0) -> a__U61(a__isNat(N)) 8.51/2.97 a__x(N, s(M)) -> a__U71(a__isNat(M), M, N) 8.51/2.97 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 8.51/2.97 mark(U12(X)) -> a__U12(mark(X)) 8.51/2.97 mark(isNat(X)) -> a__isNat(X) 8.51/2.97 mark(U21(X)) -> a__U21(mark(X)) 8.51/2.97 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 8.51/2.97 mark(U32(X)) -> a__U32(mark(X)) 8.51/2.97 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 8.51/2.97 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 8.51/2.97 mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3) 8.51/2.97 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 8.51/2.97 mark(U61(X)) -> a__U61(mark(X)) 8.51/2.97 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 8.51/2.97 mark(U72(X1, X2, X3)) -> a__U72(mark(X1), X2, X3) 8.51/2.97 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 8.51/2.97 mark(tt) -> tt 8.51/2.97 mark(s(X)) -> s(mark(X)) 8.51/2.97 mark(0) -> 0 8.51/2.97 a__U11(X1, X2) -> U11(X1, X2) 8.51/2.97 a__U12(X) -> U12(X) 8.51/2.97 a__isNat(X) -> isNat(X) 8.51/2.97 a__U21(X) -> U21(X) 8.51/2.97 a__U31(X1, X2) -> U31(X1, X2) 8.51/2.97 a__U32(X) -> U32(X) 8.51/2.97 a__U41(X1, X2) -> U41(X1, X2) 8.51/2.97 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 8.51/2.97 a__U52(X1, X2, X3) -> U52(X1, X2, X3) 8.51/2.97 a__plus(X1, X2) -> plus(X1, X2) 8.51/2.97 a__U61(X) -> U61(X) 8.51/2.97 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 8.51/2.97 a__U72(X1, X2, X3) -> U72(X1, X2, X3) 8.51/2.97 a__x(X1, X2) -> x(X1, X2) 8.51/2.97 8.51/2.97 S is empty. 8.51/2.97 Rewrite Strategy: INNERMOST 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (9) DecreasingLoopProof (FINISHED) 8.51/2.97 The following loop(s) give(s) rise to the lower bound EXP: 8.51/2.97 8.51/2.97 The rewrite sequence 8.51/2.97 8.51/2.97 mark(U72(tt, X2, X3)) ->^+ a__plus(a__x(mark(X3), mark(X2)), mark(X3)) 8.51/2.97 8.51/2.97 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 8.51/2.97 8.51/2.97 The pumping substitution is [X3 / U72(tt, X2, X3)]. 8.51/2.97 8.51/2.97 The result substitution is [ ]. 8.51/2.97 8.51/2.97 8.51/2.97 8.51/2.97 The rewrite sequence 8.51/2.97 8.51/2.97 mark(U72(tt, X2, X3)) ->^+ a__plus(a__x(mark(X3), mark(X2)), mark(X3)) 8.51/2.97 8.51/2.97 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 8.51/2.97 8.51/2.97 The pumping substitution is [X3 / U72(tt, X2, X3)]. 8.51/2.97 8.51/2.97 The result substitution is [ ]. 8.51/2.97 8.51/2.97 8.51/2.97 8.51/2.97 8.51/2.97 ---------------------------------------- 8.51/2.97 8.51/2.97 (10) 8.51/2.97 BOUNDS(EXP, INF) 8.62/3.02 EOF