5.68/2.18 WORST_CASE(NON_POLY, ?) 5.73/2.20 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.73/2.20 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.73/2.20 5.73/2.20 5.73/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.73/2.20 5.73/2.20 (0) CpxTRS 5.73/2.20 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 5.73/2.20 (2) TRS for Loop Detection 5.73/2.20 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 5.73/2.20 (4) BEST 5.73/2.20 (5) proven lower bound 5.73/2.20 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 5.73/2.20 (7) BOUNDS(n^1, INF) 5.73/2.20 (8) TRS for Loop Detection 5.73/2.20 (9) DecreasingLoopProof [FINISHED, 410 ms] 5.73/2.20 (10) BOUNDS(EXP, INF) 5.73/2.20 5.73/2.20 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (0) 5.73/2.20 Obligation: 5.73/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.73/2.20 5.73/2.20 5.73/2.20 The TRS R consists of the following rules: 5.73/2.20 5.73/2.20 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 5.73/2.20 a__U12(tt) -> tt 5.73/2.20 a__U21(tt) -> tt 5.73/2.20 a__U31(tt, N) -> mark(N) 5.73/2.20 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 5.73/2.20 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 5.73/2.20 a__isNat(0) -> tt 5.73/2.20 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 5.73/2.20 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 5.73/2.20 a__plus(N, 0) -> a__U31(a__isNat(N), N) 5.73/2.20 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 5.73/2.20 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 5.73/2.20 mark(U12(X)) -> a__U12(mark(X)) 5.73/2.20 mark(isNat(X)) -> a__isNat(X) 5.73/2.20 mark(U21(X)) -> a__U21(mark(X)) 5.73/2.20 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.73/2.20 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.73/2.20 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.73/2.20 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 5.73/2.20 mark(tt) -> tt 5.73/2.20 mark(s(X)) -> s(mark(X)) 5.73/2.20 mark(0) -> 0 5.73/2.20 a__U11(X1, X2) -> U11(X1, X2) 5.73/2.20 a__U12(X) -> U12(X) 5.73/2.20 a__isNat(X) -> isNat(X) 5.73/2.20 a__U21(X) -> U21(X) 5.73/2.20 a__U31(X1, X2) -> U31(X1, X2) 5.73/2.20 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.73/2.20 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.73/2.20 a__plus(X1, X2) -> plus(X1, X2) 5.73/2.20 5.73/2.20 S is empty. 5.73/2.20 Rewrite Strategy: FULL 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 5.73/2.20 Transformed a relative TRS into a decreasing-loop problem. 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (2) 5.73/2.20 Obligation: 5.73/2.20 Analyzing the following TRS for decreasing loops: 5.73/2.20 5.73/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.73/2.20 5.73/2.20 5.73/2.20 The TRS R consists of the following rules: 5.73/2.20 5.73/2.20 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 5.73/2.20 a__U12(tt) -> tt 5.73/2.20 a__U21(tt) -> tt 5.73/2.20 a__U31(tt, N) -> mark(N) 5.73/2.20 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 5.73/2.20 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 5.73/2.20 a__isNat(0) -> tt 5.73/2.20 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 5.73/2.20 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 5.73/2.20 a__plus(N, 0) -> a__U31(a__isNat(N), N) 5.73/2.20 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 5.73/2.20 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 5.73/2.20 mark(U12(X)) -> a__U12(mark(X)) 5.73/2.20 mark(isNat(X)) -> a__isNat(X) 5.73/2.20 mark(U21(X)) -> a__U21(mark(X)) 5.73/2.20 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.73/2.20 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.73/2.20 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.73/2.20 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 5.73/2.20 mark(tt) -> tt 5.73/2.20 mark(s(X)) -> s(mark(X)) 5.73/2.20 mark(0) -> 0 5.73/2.20 a__U11(X1, X2) -> U11(X1, X2) 5.73/2.20 a__U12(X) -> U12(X) 5.73/2.20 a__isNat(X) -> isNat(X) 5.73/2.20 a__U21(X) -> U21(X) 5.73/2.20 a__U31(X1, X2) -> U31(X1, X2) 5.73/2.20 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.73/2.20 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.73/2.20 a__plus(X1, X2) -> plus(X1, X2) 5.73/2.20 5.73/2.20 S is empty. 5.73/2.20 Rewrite Strategy: FULL 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (3) DecreasingLoopProof (LOWER BOUND(ID)) 5.73/2.20 The following loop(s) give(s) rise to the lower bound Omega(n^1): 5.73/2.20 5.73/2.20 The rewrite sequence 5.73/2.20 5.73/2.20 mark(U12(X)) ->^+ a__U12(mark(X)) 5.73/2.20 5.73/2.20 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 5.73/2.20 5.73/2.20 The pumping substitution is [X / U12(X)]. 5.73/2.20 5.73/2.20 The result substitution is [ ]. 5.73/2.20 5.73/2.20 5.73/2.20 5.73/2.20 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (4) 5.73/2.20 Complex Obligation (BEST) 5.73/2.20 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (5) 5.73/2.20 Obligation: 5.73/2.20 Proved the lower bound n^1 for the following obligation: 5.73/2.20 5.73/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.73/2.20 5.73/2.20 5.73/2.20 The TRS R consists of the following rules: 5.73/2.20 5.73/2.20 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 5.73/2.20 a__U12(tt) -> tt 5.73/2.20 a__U21(tt) -> tt 5.73/2.20 a__U31(tt, N) -> mark(N) 5.73/2.20 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 5.73/2.20 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 5.73/2.20 a__isNat(0) -> tt 5.73/2.20 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 5.73/2.20 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 5.73/2.20 a__plus(N, 0) -> a__U31(a__isNat(N), N) 5.73/2.20 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 5.73/2.20 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 5.73/2.20 mark(U12(X)) -> a__U12(mark(X)) 5.73/2.20 mark(isNat(X)) -> a__isNat(X) 5.73/2.20 mark(U21(X)) -> a__U21(mark(X)) 5.73/2.20 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.73/2.20 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.73/2.20 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.73/2.20 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 5.73/2.20 mark(tt) -> tt 5.73/2.20 mark(s(X)) -> s(mark(X)) 5.73/2.20 mark(0) -> 0 5.73/2.20 a__U11(X1, X2) -> U11(X1, X2) 5.73/2.20 a__U12(X) -> U12(X) 5.73/2.20 a__isNat(X) -> isNat(X) 5.73/2.20 a__U21(X) -> U21(X) 5.73/2.20 a__U31(X1, X2) -> U31(X1, X2) 5.73/2.20 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.73/2.20 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.73/2.20 a__plus(X1, X2) -> plus(X1, X2) 5.73/2.20 5.73/2.20 S is empty. 5.73/2.20 Rewrite Strategy: FULL 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (6) LowerBoundPropagationProof (FINISHED) 5.73/2.20 Propagated lower bound. 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (7) 5.73/2.20 BOUNDS(n^1, INF) 5.73/2.20 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (8) 5.73/2.20 Obligation: 5.73/2.20 Analyzing the following TRS for decreasing loops: 5.73/2.20 5.73/2.20 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.73/2.20 5.73/2.20 5.73/2.20 The TRS R consists of the following rules: 5.73/2.20 5.73/2.20 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 5.73/2.20 a__U12(tt) -> tt 5.73/2.20 a__U21(tt) -> tt 5.73/2.20 a__U31(tt, N) -> mark(N) 5.73/2.20 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 5.73/2.20 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 5.73/2.20 a__isNat(0) -> tt 5.73/2.20 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 5.73/2.20 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 5.73/2.20 a__plus(N, 0) -> a__U31(a__isNat(N), N) 5.73/2.20 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 5.73/2.20 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 5.73/2.20 mark(U12(X)) -> a__U12(mark(X)) 5.73/2.20 mark(isNat(X)) -> a__isNat(X) 5.73/2.20 mark(U21(X)) -> a__U21(mark(X)) 5.73/2.20 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.73/2.20 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.73/2.20 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.73/2.20 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 5.73/2.20 mark(tt) -> tt 5.73/2.20 mark(s(X)) -> s(mark(X)) 5.73/2.20 mark(0) -> 0 5.73/2.20 a__U11(X1, X2) -> U11(X1, X2) 5.73/2.20 a__U12(X) -> U12(X) 5.73/2.20 a__isNat(X) -> isNat(X) 5.73/2.20 a__U21(X) -> U21(X) 5.73/2.20 a__U31(X1, X2) -> U31(X1, X2) 5.73/2.20 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.73/2.20 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.73/2.20 a__plus(X1, X2) -> plus(X1, X2) 5.73/2.20 5.73/2.20 S is empty. 5.73/2.20 Rewrite Strategy: FULL 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (9) DecreasingLoopProof (FINISHED) 5.73/2.20 The following loop(s) give(s) rise to the lower bound EXP: 5.73/2.20 5.73/2.20 The rewrite sequence 5.73/2.20 5.73/2.20 mark(plus(X1, s(X1_0))) ->^+ a__U41(a__isNat(mark(X1_0)), mark(X1_0), mark(X1)) 5.73/2.20 5.73/2.20 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 5.73/2.20 5.73/2.20 The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. 5.73/2.20 5.73/2.20 The result substitution is [ ]. 5.73/2.20 5.73/2.20 5.73/2.20 5.73/2.20 The rewrite sequence 5.73/2.20 5.73/2.20 mark(plus(X1, s(X1_0))) ->^+ a__U41(a__isNat(mark(X1_0)), mark(X1_0), mark(X1)) 5.73/2.20 5.73/2.20 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 5.73/2.20 5.73/2.20 The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. 5.73/2.20 5.73/2.20 The result substitution is [ ]. 5.73/2.20 5.73/2.20 5.73/2.20 5.73/2.20 5.73/2.20 ---------------------------------------- 5.73/2.20 5.73/2.20 (10) 5.73/2.20 BOUNDS(EXP, INF) 5.77/2.23 EOF