340.73/291.52 WORST_CASE(Omega(n^1), ?) 340.73/291.53 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 340.73/291.53 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 340.73/291.53 340.73/291.53 340.73/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 340.73/291.53 340.73/291.53 (0) CpxTRS 340.73/291.53 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 340.73/291.53 (2) TRS for Loop Detection 340.73/291.53 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 340.73/291.53 (4) BEST 340.73/291.53 (5) proven lower bound 340.73/291.53 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 340.73/291.53 (7) BOUNDS(n^1, INF) 340.73/291.53 (8) TRS for Loop Detection 340.73/291.53 340.73/291.53 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (0) 340.73/291.53 Obligation: 340.73/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 340.73/291.53 340.73/291.53 340.73/291.53 The TRS R consists of the following rules: 340.73/291.53 340.73/291.53 a__dbl(0) -> 0 340.73/291.53 a__dbl(s(X)) -> s(s(dbl(X))) 340.73/291.53 a__dbls(nil) -> nil 340.73/291.53 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 340.73/291.53 a__sel(0, cons(X, Y)) -> mark(X) 340.73/291.53 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 340.73/291.53 a__indx(nil, X) -> nil 340.73/291.53 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 340.73/291.53 a__from(X) -> cons(X, from(s(X))) 340.73/291.53 mark(dbl(X)) -> a__dbl(mark(X)) 340.73/291.53 mark(dbls(X)) -> a__dbls(mark(X)) 340.73/291.53 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 340.73/291.53 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 340.73/291.53 mark(from(X)) -> a__from(X) 340.73/291.53 mark(0) -> 0 340.73/291.53 mark(s(X)) -> s(X) 340.73/291.53 mark(nil) -> nil 340.73/291.53 mark(cons(X1, X2)) -> cons(X1, X2) 340.73/291.53 a__dbl(X) -> dbl(X) 340.73/291.53 a__dbls(X) -> dbls(X) 340.73/291.53 a__sel(X1, X2) -> sel(X1, X2) 340.73/291.53 a__indx(X1, X2) -> indx(X1, X2) 340.73/291.53 a__from(X) -> from(X) 340.73/291.53 340.73/291.53 S is empty. 340.73/291.53 Rewrite Strategy: FULL 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 340.73/291.53 Transformed a relative TRS into a decreasing-loop problem. 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (2) 340.73/291.53 Obligation: 340.73/291.53 Analyzing the following TRS for decreasing loops: 340.73/291.53 340.73/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 340.73/291.53 340.73/291.53 340.73/291.53 The TRS R consists of the following rules: 340.73/291.53 340.73/291.53 a__dbl(0) -> 0 340.73/291.53 a__dbl(s(X)) -> s(s(dbl(X))) 340.73/291.53 a__dbls(nil) -> nil 340.73/291.53 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 340.73/291.53 a__sel(0, cons(X, Y)) -> mark(X) 340.73/291.53 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 340.73/291.53 a__indx(nil, X) -> nil 340.73/291.53 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 340.73/291.53 a__from(X) -> cons(X, from(s(X))) 340.73/291.53 mark(dbl(X)) -> a__dbl(mark(X)) 340.73/291.53 mark(dbls(X)) -> a__dbls(mark(X)) 340.73/291.53 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 340.73/291.53 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 340.73/291.53 mark(from(X)) -> a__from(X) 340.73/291.53 mark(0) -> 0 340.73/291.53 mark(s(X)) -> s(X) 340.73/291.53 mark(nil) -> nil 340.73/291.53 mark(cons(X1, X2)) -> cons(X1, X2) 340.73/291.53 a__dbl(X) -> dbl(X) 340.73/291.53 a__dbls(X) -> dbls(X) 340.73/291.53 a__sel(X1, X2) -> sel(X1, X2) 340.73/291.53 a__indx(X1, X2) -> indx(X1, X2) 340.73/291.53 a__from(X) -> from(X) 340.73/291.53 340.73/291.53 S is empty. 340.73/291.53 Rewrite Strategy: FULL 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (3) DecreasingLoopProof (LOWER BOUND(ID)) 340.73/291.53 The following loop(s) give(s) rise to the lower bound Omega(n^1): 340.73/291.53 340.73/291.53 The rewrite sequence 340.73/291.53 340.73/291.53 mark(indx(X1, X2)) ->^+ a__indx(mark(X1), X2) 340.73/291.53 340.73/291.53 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 340.73/291.53 340.73/291.53 The pumping substitution is [X1 / indx(X1, X2)]. 340.73/291.53 340.73/291.53 The result substitution is [ ]. 340.73/291.53 340.73/291.53 340.73/291.53 340.73/291.53 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (4) 340.73/291.53 Complex Obligation (BEST) 340.73/291.53 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (5) 340.73/291.53 Obligation: 340.73/291.53 Proved the lower bound n^1 for the following obligation: 340.73/291.53 340.73/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 340.73/291.53 340.73/291.53 340.73/291.53 The TRS R consists of the following rules: 340.73/291.53 340.73/291.53 a__dbl(0) -> 0 340.73/291.53 a__dbl(s(X)) -> s(s(dbl(X))) 340.73/291.53 a__dbls(nil) -> nil 340.73/291.53 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 340.73/291.53 a__sel(0, cons(X, Y)) -> mark(X) 340.73/291.53 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 340.73/291.53 a__indx(nil, X) -> nil 340.73/291.53 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 340.73/291.53 a__from(X) -> cons(X, from(s(X))) 340.73/291.53 mark(dbl(X)) -> a__dbl(mark(X)) 340.73/291.53 mark(dbls(X)) -> a__dbls(mark(X)) 340.73/291.53 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 340.73/291.53 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 340.73/291.53 mark(from(X)) -> a__from(X) 340.73/291.53 mark(0) -> 0 340.73/291.53 mark(s(X)) -> s(X) 340.73/291.53 mark(nil) -> nil 340.73/291.53 mark(cons(X1, X2)) -> cons(X1, X2) 340.73/291.53 a__dbl(X) -> dbl(X) 340.73/291.53 a__dbls(X) -> dbls(X) 340.73/291.53 a__sel(X1, X2) -> sel(X1, X2) 340.73/291.53 a__indx(X1, X2) -> indx(X1, X2) 340.73/291.53 a__from(X) -> from(X) 340.73/291.53 340.73/291.53 S is empty. 340.73/291.53 Rewrite Strategy: FULL 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (6) LowerBoundPropagationProof (FINISHED) 340.73/291.53 Propagated lower bound. 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (7) 340.73/291.53 BOUNDS(n^1, INF) 340.73/291.53 340.73/291.53 ---------------------------------------- 340.73/291.53 340.73/291.53 (8) 340.73/291.53 Obligation: 340.73/291.53 Analyzing the following TRS for decreasing loops: 340.73/291.53 340.73/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 340.73/291.53 340.73/291.53 340.73/291.53 The TRS R consists of the following rules: 340.73/291.53 340.73/291.53 a__dbl(0) -> 0 340.73/291.53 a__dbl(s(X)) -> s(s(dbl(X))) 340.73/291.53 a__dbls(nil) -> nil 340.73/291.53 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 340.73/291.53 a__sel(0, cons(X, Y)) -> mark(X) 340.73/291.53 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 340.73/291.53 a__indx(nil, X) -> nil 340.73/291.53 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 340.73/291.53 a__from(X) -> cons(X, from(s(X))) 340.73/291.53 mark(dbl(X)) -> a__dbl(mark(X)) 340.73/291.53 mark(dbls(X)) -> a__dbls(mark(X)) 340.73/291.53 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 340.73/291.53 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 340.73/291.53 mark(from(X)) -> a__from(X) 340.73/291.53 mark(0) -> 0 340.73/291.53 mark(s(X)) -> s(X) 340.73/291.53 mark(nil) -> nil 340.73/291.53 mark(cons(X1, X2)) -> cons(X1, X2) 340.73/291.53 a__dbl(X) -> dbl(X) 340.73/291.53 a__dbls(X) -> dbls(X) 340.73/291.53 a__sel(X1, X2) -> sel(X1, X2) 340.73/291.53 a__indx(X1, X2) -> indx(X1, X2) 340.73/291.53 a__from(X) -> from(X) 340.73/291.53 340.73/291.53 S is empty. 340.73/291.53 Rewrite Strategy: FULL 340.73/291.56 EOF