490.32/291.56 WORST_CASE(Omega(n^1), ?) 490.37/291.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 490.37/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 490.37/291.57 490.37/291.57 490.37/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 490.37/291.57 490.37/291.57 (0) CpxTRS 490.37/291.57 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 490.37/291.57 (2) TRS for Loop Detection 490.37/291.57 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 490.37/291.57 (4) BEST 490.37/291.57 (5) proven lower bound 490.37/291.57 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 490.37/291.57 (7) BOUNDS(n^1, INF) 490.37/291.57 (8) TRS for Loop Detection 490.37/291.57 490.37/291.57 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (0) 490.37/291.57 Obligation: 490.37/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 490.37/291.57 490.37/291.57 490.37/291.57 The TRS R consists of the following rules: 490.37/291.57 490.37/291.57 a__dbl(0) -> 0 490.37/291.57 a__dbl(s(X)) -> s(s(dbl(X))) 490.37/291.57 a__dbls(nil) -> nil 490.37/291.57 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 490.37/291.57 a__sel(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 490.37/291.57 a__indx(nil, X) -> nil 490.37/291.57 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 490.37/291.57 a__from(X) -> cons(X, from(s(X))) 490.37/291.57 a__dbl1(0) -> 01 490.37/291.57 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 490.37/291.57 a__sel1(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 490.37/291.57 a__quote(0) -> 01 490.37/291.57 a__quote(s(X)) -> s1(a__quote(mark(X))) 490.37/291.57 a__quote(dbl(X)) -> a__dbl1(mark(X)) 490.37/291.57 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 490.37/291.57 mark(dbl(X)) -> a__dbl(mark(X)) 490.37/291.57 mark(dbls(X)) -> a__dbls(mark(X)) 490.37/291.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 490.37/291.57 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 490.37/291.57 mark(from(X)) -> a__from(X) 490.37/291.57 mark(dbl1(X)) -> a__dbl1(mark(X)) 490.37/291.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 490.37/291.57 mark(quote(X)) -> a__quote(mark(X)) 490.37/291.57 mark(0) -> 0 490.37/291.57 mark(s(X)) -> s(X) 490.37/291.57 mark(nil) -> nil 490.37/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 490.37/291.57 mark(01) -> 01 490.37/291.57 mark(s1(X)) -> s1(mark(X)) 490.37/291.57 a__dbl(X) -> dbl(X) 490.37/291.57 a__dbls(X) -> dbls(X) 490.37/291.57 a__sel(X1, X2) -> sel(X1, X2) 490.37/291.57 a__indx(X1, X2) -> indx(X1, X2) 490.37/291.57 a__from(X) -> from(X) 490.37/291.57 a__dbl1(X) -> dbl1(X) 490.37/291.57 a__sel1(X1, X2) -> sel1(X1, X2) 490.37/291.57 a__quote(X) -> quote(X) 490.37/291.57 490.37/291.57 S is empty. 490.37/291.57 Rewrite Strategy: FULL 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 490.37/291.57 Transformed a relative TRS into a decreasing-loop problem. 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (2) 490.37/291.57 Obligation: 490.37/291.57 Analyzing the following TRS for decreasing loops: 490.37/291.57 490.37/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 490.37/291.57 490.37/291.57 490.37/291.57 The TRS R consists of the following rules: 490.37/291.57 490.37/291.57 a__dbl(0) -> 0 490.37/291.57 a__dbl(s(X)) -> s(s(dbl(X))) 490.37/291.57 a__dbls(nil) -> nil 490.37/291.57 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 490.37/291.57 a__sel(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 490.37/291.57 a__indx(nil, X) -> nil 490.37/291.57 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 490.37/291.57 a__from(X) -> cons(X, from(s(X))) 490.37/291.57 a__dbl1(0) -> 01 490.37/291.57 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 490.37/291.57 a__sel1(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 490.37/291.57 a__quote(0) -> 01 490.37/291.57 a__quote(s(X)) -> s1(a__quote(mark(X))) 490.37/291.57 a__quote(dbl(X)) -> a__dbl1(mark(X)) 490.37/291.57 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 490.37/291.57 mark(dbl(X)) -> a__dbl(mark(X)) 490.37/291.57 mark(dbls(X)) -> a__dbls(mark(X)) 490.37/291.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 490.37/291.57 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 490.37/291.57 mark(from(X)) -> a__from(X) 490.37/291.57 mark(dbl1(X)) -> a__dbl1(mark(X)) 490.37/291.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 490.37/291.57 mark(quote(X)) -> a__quote(mark(X)) 490.37/291.57 mark(0) -> 0 490.37/291.57 mark(s(X)) -> s(X) 490.37/291.57 mark(nil) -> nil 490.37/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 490.37/291.57 mark(01) -> 01 490.37/291.57 mark(s1(X)) -> s1(mark(X)) 490.37/291.57 a__dbl(X) -> dbl(X) 490.37/291.57 a__dbls(X) -> dbls(X) 490.37/291.57 a__sel(X1, X2) -> sel(X1, X2) 490.37/291.57 a__indx(X1, X2) -> indx(X1, X2) 490.37/291.57 a__from(X) -> from(X) 490.37/291.57 a__dbl1(X) -> dbl1(X) 490.37/291.57 a__sel1(X1, X2) -> sel1(X1, X2) 490.37/291.57 a__quote(X) -> quote(X) 490.37/291.57 490.37/291.57 S is empty. 490.37/291.57 Rewrite Strategy: FULL 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (3) DecreasingLoopProof (LOWER BOUND(ID)) 490.37/291.57 The following loop(s) give(s) rise to the lower bound Omega(n^1): 490.37/291.57 490.37/291.57 The rewrite sequence 490.37/291.57 490.37/291.57 mark(indx(X1, X2)) ->^+ a__indx(mark(X1), X2) 490.37/291.57 490.37/291.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 490.37/291.57 490.37/291.57 The pumping substitution is [X1 / indx(X1, X2)]. 490.37/291.57 490.37/291.57 The result substitution is [ ]. 490.37/291.57 490.37/291.57 490.37/291.57 490.37/291.57 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (4) 490.37/291.57 Complex Obligation (BEST) 490.37/291.57 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (5) 490.37/291.57 Obligation: 490.37/291.57 Proved the lower bound n^1 for the following obligation: 490.37/291.57 490.37/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 490.37/291.57 490.37/291.57 490.37/291.57 The TRS R consists of the following rules: 490.37/291.57 490.37/291.57 a__dbl(0) -> 0 490.37/291.57 a__dbl(s(X)) -> s(s(dbl(X))) 490.37/291.57 a__dbls(nil) -> nil 490.37/291.57 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 490.37/291.57 a__sel(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 490.37/291.57 a__indx(nil, X) -> nil 490.37/291.57 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 490.37/291.57 a__from(X) -> cons(X, from(s(X))) 490.37/291.57 a__dbl1(0) -> 01 490.37/291.57 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 490.37/291.57 a__sel1(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 490.37/291.57 a__quote(0) -> 01 490.37/291.57 a__quote(s(X)) -> s1(a__quote(mark(X))) 490.37/291.57 a__quote(dbl(X)) -> a__dbl1(mark(X)) 490.37/291.57 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 490.37/291.57 mark(dbl(X)) -> a__dbl(mark(X)) 490.37/291.57 mark(dbls(X)) -> a__dbls(mark(X)) 490.37/291.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 490.37/291.57 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 490.37/291.57 mark(from(X)) -> a__from(X) 490.37/291.57 mark(dbl1(X)) -> a__dbl1(mark(X)) 490.37/291.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 490.37/291.57 mark(quote(X)) -> a__quote(mark(X)) 490.37/291.57 mark(0) -> 0 490.37/291.57 mark(s(X)) -> s(X) 490.37/291.57 mark(nil) -> nil 490.37/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 490.37/291.57 mark(01) -> 01 490.37/291.57 mark(s1(X)) -> s1(mark(X)) 490.37/291.57 a__dbl(X) -> dbl(X) 490.37/291.57 a__dbls(X) -> dbls(X) 490.37/291.57 a__sel(X1, X2) -> sel(X1, X2) 490.37/291.57 a__indx(X1, X2) -> indx(X1, X2) 490.37/291.57 a__from(X) -> from(X) 490.37/291.57 a__dbl1(X) -> dbl1(X) 490.37/291.57 a__sel1(X1, X2) -> sel1(X1, X2) 490.37/291.57 a__quote(X) -> quote(X) 490.37/291.57 490.37/291.57 S is empty. 490.37/291.57 Rewrite Strategy: FULL 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (6) LowerBoundPropagationProof (FINISHED) 490.37/291.57 Propagated lower bound. 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (7) 490.37/291.57 BOUNDS(n^1, INF) 490.37/291.57 490.37/291.57 ---------------------------------------- 490.37/291.57 490.37/291.57 (8) 490.37/291.57 Obligation: 490.37/291.57 Analyzing the following TRS for decreasing loops: 490.37/291.57 490.37/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 490.37/291.57 490.37/291.57 490.37/291.57 The TRS R consists of the following rules: 490.37/291.57 490.37/291.57 a__dbl(0) -> 0 490.37/291.57 a__dbl(s(X)) -> s(s(dbl(X))) 490.37/291.57 a__dbls(nil) -> nil 490.37/291.57 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 490.37/291.57 a__sel(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 490.37/291.57 a__indx(nil, X) -> nil 490.37/291.57 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 490.37/291.57 a__from(X) -> cons(X, from(s(X))) 490.37/291.57 a__dbl1(0) -> 01 490.37/291.57 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 490.37/291.57 a__sel1(0, cons(X, Y)) -> mark(X) 490.37/291.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 490.37/291.57 a__quote(0) -> 01 490.37/291.57 a__quote(s(X)) -> s1(a__quote(mark(X))) 490.37/291.57 a__quote(dbl(X)) -> a__dbl1(mark(X)) 490.37/291.57 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 490.37/291.57 mark(dbl(X)) -> a__dbl(mark(X)) 490.37/291.57 mark(dbls(X)) -> a__dbls(mark(X)) 490.37/291.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 490.37/291.57 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 490.37/291.57 mark(from(X)) -> a__from(X) 490.37/291.57 mark(dbl1(X)) -> a__dbl1(mark(X)) 490.37/291.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 490.37/291.57 mark(quote(X)) -> a__quote(mark(X)) 490.37/291.57 mark(0) -> 0 490.37/291.57 mark(s(X)) -> s(X) 490.37/291.57 mark(nil) -> nil 490.37/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 490.37/291.57 mark(01) -> 01 490.37/291.57 mark(s1(X)) -> s1(mark(X)) 490.37/291.57 a__dbl(X) -> dbl(X) 490.37/291.57 a__dbls(X) -> dbls(X) 490.37/291.57 a__sel(X1, X2) -> sel(X1, X2) 490.37/291.57 a__indx(X1, X2) -> indx(X1, X2) 490.37/291.57 a__from(X) -> from(X) 490.37/291.57 a__dbl1(X) -> dbl1(X) 490.37/291.57 a__sel1(X1, X2) -> sel1(X1, X2) 490.37/291.57 a__quote(X) -> quote(X) 490.37/291.57 490.37/291.57 S is empty. 490.37/291.57 Rewrite Strategy: FULL 490.37/291.62 EOF