1104.88/291.51 WORST_CASE(Omega(n^1), ?) 1104.88/291.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1104.88/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1104.88/291.51 1104.88/291.51 1104.88/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.88/291.51 1104.88/291.51 (0) CpxTRS 1104.88/291.51 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1104.88/291.51 (2) TRS for Loop Detection 1104.88/291.51 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1104.88/291.51 (4) BEST 1104.88/291.51 (5) proven lower bound 1104.88/291.51 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1104.88/291.51 (7) BOUNDS(n^1, INF) 1104.88/291.51 (8) TRS for Loop Detection 1104.88/291.51 1104.88/291.51 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (0) 1104.88/291.51 Obligation: 1104.88/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.88/291.51 1104.88/291.51 1104.88/291.51 The TRS R consists of the following rules: 1104.88/291.51 1104.88/291.51 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 1104.88/291.51 a____(X, nil) -> mark(X) 1104.88/291.51 a____(nil, X) -> mark(X) 1104.88/291.51 a__U11(tt) -> a__U12(tt) 1104.88/291.51 a__U12(tt) -> tt 1104.88/291.51 a__isNePal(__(I, __(P, I))) -> a__U11(tt) 1104.88/291.51 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 1104.88/291.51 mark(U11(X)) -> a__U11(mark(X)) 1104.88/291.51 mark(U12(X)) -> a__U12(mark(X)) 1104.88/291.51 mark(isNePal(X)) -> a__isNePal(mark(X)) 1104.88/291.51 mark(nil) -> nil 1104.88/291.51 mark(tt) -> tt 1104.88/291.51 a____(X1, X2) -> __(X1, X2) 1104.88/291.51 a__U11(X) -> U11(X) 1104.88/291.51 a__U12(X) -> U12(X) 1104.88/291.51 a__isNePal(X) -> isNePal(X) 1104.88/291.51 1104.88/291.51 S is empty. 1104.88/291.51 Rewrite Strategy: FULL 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1104.88/291.51 Transformed a relative TRS into a decreasing-loop problem. 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (2) 1104.88/291.51 Obligation: 1104.88/291.51 Analyzing the following TRS for decreasing loops: 1104.88/291.51 1104.88/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.88/291.51 1104.88/291.51 1104.88/291.51 The TRS R consists of the following rules: 1104.88/291.51 1104.88/291.51 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 1104.88/291.51 a____(X, nil) -> mark(X) 1104.88/291.51 a____(nil, X) -> mark(X) 1104.88/291.51 a__U11(tt) -> a__U12(tt) 1104.88/291.51 a__U12(tt) -> tt 1104.88/291.51 a__isNePal(__(I, __(P, I))) -> a__U11(tt) 1104.88/291.51 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 1104.88/291.51 mark(U11(X)) -> a__U11(mark(X)) 1104.88/291.51 mark(U12(X)) -> a__U12(mark(X)) 1104.88/291.51 mark(isNePal(X)) -> a__isNePal(mark(X)) 1104.88/291.51 mark(nil) -> nil 1104.88/291.51 mark(tt) -> tt 1104.88/291.51 a____(X1, X2) -> __(X1, X2) 1104.88/291.51 a__U11(X) -> U11(X) 1104.88/291.51 a__U12(X) -> U12(X) 1104.88/291.51 a__isNePal(X) -> isNePal(X) 1104.88/291.51 1104.88/291.51 S is empty. 1104.88/291.51 Rewrite Strategy: FULL 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1104.88/291.51 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1104.88/291.51 1104.88/291.51 The rewrite sequence 1104.88/291.51 1104.88/291.51 mark(U12(X)) ->^+ a__U12(mark(X)) 1104.88/291.51 1104.88/291.51 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1104.88/291.51 1104.88/291.51 The pumping substitution is [X / U12(X)]. 1104.88/291.51 1104.88/291.51 The result substitution is [ ]. 1104.88/291.51 1104.88/291.51 1104.88/291.51 1104.88/291.51 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (4) 1104.88/291.51 Complex Obligation (BEST) 1104.88/291.51 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (5) 1104.88/291.51 Obligation: 1104.88/291.51 Proved the lower bound n^1 for the following obligation: 1104.88/291.51 1104.88/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.88/291.51 1104.88/291.51 1104.88/291.51 The TRS R consists of the following rules: 1104.88/291.51 1104.88/291.51 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 1104.88/291.51 a____(X, nil) -> mark(X) 1104.88/291.51 a____(nil, X) -> mark(X) 1104.88/291.51 a__U11(tt) -> a__U12(tt) 1104.88/291.51 a__U12(tt) -> tt 1104.88/291.51 a__isNePal(__(I, __(P, I))) -> a__U11(tt) 1104.88/291.51 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 1104.88/291.51 mark(U11(X)) -> a__U11(mark(X)) 1104.88/291.51 mark(U12(X)) -> a__U12(mark(X)) 1104.88/291.51 mark(isNePal(X)) -> a__isNePal(mark(X)) 1104.88/291.51 mark(nil) -> nil 1104.88/291.51 mark(tt) -> tt 1104.88/291.51 a____(X1, X2) -> __(X1, X2) 1104.88/291.51 a__U11(X) -> U11(X) 1104.88/291.51 a__U12(X) -> U12(X) 1104.88/291.51 a__isNePal(X) -> isNePal(X) 1104.88/291.51 1104.88/291.51 S is empty. 1104.88/291.51 Rewrite Strategy: FULL 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (6) LowerBoundPropagationProof (FINISHED) 1104.88/291.51 Propagated lower bound. 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (7) 1104.88/291.51 BOUNDS(n^1, INF) 1104.88/291.51 1104.88/291.51 ---------------------------------------- 1104.88/291.51 1104.88/291.51 (8) 1104.88/291.51 Obligation: 1104.88/291.51 Analyzing the following TRS for decreasing loops: 1104.88/291.51 1104.88/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.88/291.51 1104.88/291.51 1104.88/291.51 The TRS R consists of the following rules: 1104.88/291.51 1104.88/291.51 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 1104.88/291.51 a____(X, nil) -> mark(X) 1104.88/291.51 a____(nil, X) -> mark(X) 1104.88/291.51 a__U11(tt) -> a__U12(tt) 1104.88/291.51 a__U12(tt) -> tt 1104.88/291.51 a__isNePal(__(I, __(P, I))) -> a__U11(tt) 1104.88/291.51 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 1104.88/291.51 mark(U11(X)) -> a__U11(mark(X)) 1104.88/291.51 mark(U12(X)) -> a__U12(mark(X)) 1104.88/291.51 mark(isNePal(X)) -> a__isNePal(mark(X)) 1104.88/291.51 mark(nil) -> nil 1104.88/291.51 mark(tt) -> tt 1104.88/291.51 a____(X1, X2) -> __(X1, X2) 1104.88/291.51 a__U11(X) -> U11(X) 1104.88/291.51 a__U12(X) -> U12(X) 1104.88/291.51 a__isNePal(X) -> isNePal(X) 1104.88/291.51 1104.88/291.51 S is empty. 1104.88/291.51 Rewrite Strategy: FULL 1104.94/291.57 EOF