1117.69/292.27 WORST_CASE(Omega(n^1), ?) 1120.91/293.14 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1120.91/293.14 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1120.91/293.14 1120.91/293.14 1120.91/293.14 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1120.91/293.14 1120.91/293.14 (0) CpxTRS 1120.91/293.14 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1120.91/293.14 (2) TRS for Loop Detection 1120.91/293.14 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1120.91/293.14 (4) BEST 1120.91/293.14 (5) proven lower bound 1120.91/293.14 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1120.91/293.14 (7) BOUNDS(n^1, INF) 1120.91/293.14 (8) TRS for Loop Detection 1120.91/293.14 1120.91/293.14 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (0) 1120.91/293.14 Obligation: 1120.91/293.14 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1120.91/293.14 1120.91/293.14 1120.91/293.14 The TRS R consists of the following rules: 1120.91/293.14 1120.91/293.14 a__U11(tt, N) -> mark(N) 1120.91/293.14 a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) 1120.91/293.14 a__and(tt, X) -> mark(X) 1120.91/293.14 a__isNat(0) -> tt 1120.91/293.14 a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) 1120.91/293.14 a__isNat(s(V1)) -> a__isNat(V1) 1120.91/293.14 a__plus(N, 0) -> a__U11(a__isNat(N), N) 1120.91/293.14 a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) 1120.91/293.14 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1120.91/293.14 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1120.91/293.14 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1120.91/293.14 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1120.91/293.14 mark(isNat(X)) -> a__isNat(X) 1120.91/293.14 mark(tt) -> tt 1120.91/293.14 mark(s(X)) -> s(mark(X)) 1120.91/293.14 mark(0) -> 0 1120.91/293.14 a__U11(X1, X2) -> U11(X1, X2) 1120.91/293.14 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1120.91/293.14 a__plus(X1, X2) -> plus(X1, X2) 1120.91/293.14 a__and(X1, X2) -> and(X1, X2) 1120.91/293.14 a__isNat(X) -> isNat(X) 1120.91/293.14 1120.91/293.14 S is empty. 1120.91/293.14 Rewrite Strategy: INNERMOST 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1120.91/293.14 Transformed a relative TRS into a decreasing-loop problem. 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (2) 1120.91/293.14 Obligation: 1120.91/293.14 Analyzing the following TRS for decreasing loops: 1120.91/293.14 1120.91/293.14 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1120.91/293.14 1120.91/293.14 1120.91/293.14 The TRS R consists of the following rules: 1120.91/293.14 1120.91/293.14 a__U11(tt, N) -> mark(N) 1120.91/293.14 a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) 1120.91/293.14 a__and(tt, X) -> mark(X) 1120.91/293.14 a__isNat(0) -> tt 1120.91/293.14 a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) 1120.91/293.14 a__isNat(s(V1)) -> a__isNat(V1) 1120.91/293.14 a__plus(N, 0) -> a__U11(a__isNat(N), N) 1120.91/293.14 a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) 1120.91/293.14 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1120.91/293.14 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1120.91/293.14 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1120.91/293.14 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1120.91/293.14 mark(isNat(X)) -> a__isNat(X) 1120.91/293.14 mark(tt) -> tt 1120.91/293.14 mark(s(X)) -> s(mark(X)) 1120.91/293.14 mark(0) -> 0 1120.91/293.14 a__U11(X1, X2) -> U11(X1, X2) 1120.91/293.14 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1120.91/293.14 a__plus(X1, X2) -> plus(X1, X2) 1120.91/293.14 a__and(X1, X2) -> and(X1, X2) 1120.91/293.14 a__isNat(X) -> isNat(X) 1120.91/293.14 1120.91/293.14 S is empty. 1120.91/293.14 Rewrite Strategy: INNERMOST 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1120.91/293.14 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1120.91/293.14 1120.91/293.14 The rewrite sequence 1120.91/293.14 1120.91/293.14 mark(U11(X1, X2)) ->^+ a__U11(mark(X1), X2) 1120.91/293.14 1120.91/293.14 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1120.91/293.14 1120.91/293.14 The pumping substitution is [X1 / U11(X1, X2)]. 1120.91/293.14 1120.91/293.14 The result substitution is [ ]. 1120.91/293.14 1120.91/293.14 1120.91/293.14 1120.91/293.14 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (4) 1120.91/293.14 Complex Obligation (BEST) 1120.91/293.14 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (5) 1120.91/293.14 Obligation: 1120.91/293.14 Proved the lower bound n^1 for the following obligation: 1120.91/293.14 1120.91/293.14 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1120.91/293.14 1120.91/293.14 1120.91/293.14 The TRS R consists of the following rules: 1120.91/293.14 1120.91/293.14 a__U11(tt, N) -> mark(N) 1120.91/293.14 a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) 1120.91/293.14 a__and(tt, X) -> mark(X) 1120.91/293.14 a__isNat(0) -> tt 1120.91/293.14 a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) 1120.91/293.14 a__isNat(s(V1)) -> a__isNat(V1) 1120.91/293.14 a__plus(N, 0) -> a__U11(a__isNat(N), N) 1120.91/293.14 a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) 1120.91/293.14 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1120.91/293.14 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1120.91/293.14 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1120.91/293.14 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1120.91/293.14 mark(isNat(X)) -> a__isNat(X) 1120.91/293.14 mark(tt) -> tt 1120.91/293.14 mark(s(X)) -> s(mark(X)) 1120.91/293.14 mark(0) -> 0 1120.91/293.14 a__U11(X1, X2) -> U11(X1, X2) 1120.91/293.14 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1120.91/293.14 a__plus(X1, X2) -> plus(X1, X2) 1120.91/293.14 a__and(X1, X2) -> and(X1, X2) 1120.91/293.14 a__isNat(X) -> isNat(X) 1120.91/293.14 1120.91/293.14 S is empty. 1120.91/293.14 Rewrite Strategy: INNERMOST 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (6) LowerBoundPropagationProof (FINISHED) 1120.91/293.14 Propagated lower bound. 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (7) 1120.91/293.14 BOUNDS(n^1, INF) 1120.91/293.14 1120.91/293.14 ---------------------------------------- 1120.91/293.14 1120.91/293.14 (8) 1120.91/293.14 Obligation: 1120.91/293.14 Analyzing the following TRS for decreasing loops: 1120.91/293.14 1120.91/293.14 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1120.91/293.14 1120.91/293.14 1120.91/293.14 The TRS R consists of the following rules: 1120.91/293.14 1120.91/293.14 a__U11(tt, N) -> mark(N) 1120.91/293.14 a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) 1120.91/293.14 a__and(tt, X) -> mark(X) 1120.91/293.14 a__isNat(0) -> tt 1120.91/293.14 a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) 1120.91/293.14 a__isNat(s(V1)) -> a__isNat(V1) 1120.91/293.14 a__plus(N, 0) -> a__U11(a__isNat(N), N) 1120.91/293.14 a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) 1120.91/293.14 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1120.91/293.14 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 1120.91/293.14 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1120.91/293.14 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1120.91/293.14 mark(isNat(X)) -> a__isNat(X) 1120.91/293.14 mark(tt) -> tt 1120.91/293.14 mark(s(X)) -> s(mark(X)) 1120.91/293.14 mark(0) -> 0 1120.91/293.14 a__U11(X1, X2) -> U11(X1, X2) 1120.91/293.14 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 1120.91/293.14 a__plus(X1, X2) -> plus(X1, X2) 1120.91/293.14 a__and(X1, X2) -> and(X1, X2) 1120.91/293.14 a__isNat(X) -> isNat(X) 1120.91/293.14 1120.91/293.14 S is empty. 1120.91/293.14 Rewrite Strategy: INNERMOST 1121.23/293.21 EOF