1140.86/291.59 WORST_CASE(Omega(n^1), ?) 1148.39/293.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1148.39/293.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1148.39/293.48 1148.39/293.48 1148.39/293.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1148.39/293.48 1148.39/293.48 (0) CpxTRS 1148.39/293.48 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1148.39/293.48 (2) TRS for Loop Detection 1148.39/293.48 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1148.39/293.48 (4) BEST 1148.39/293.48 (5) proven lower bound 1148.39/293.48 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1148.39/293.48 (7) BOUNDS(n^1, INF) 1148.39/293.48 (8) TRS for Loop Detection 1148.39/293.48 1148.39/293.48 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (0) 1148.39/293.48 Obligation: 1148.39/293.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1148.39/293.48 1148.39/293.48 1148.39/293.48 The TRS R consists of the following rules: 1148.39/293.48 1148.39/293.48 a__dbl(0) -> 0 1148.39/293.48 a__dbl(s(X)) -> s(s(dbl(X))) 1148.39/293.48 a__dbls(nil) -> nil 1148.39/293.48 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 1148.39/293.48 a__sel(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1148.39/293.48 a__indx(nil, X) -> nil 1148.39/293.48 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 1148.39/293.48 a__from(X) -> cons(X, from(s(X))) 1148.39/293.48 a__dbl1(0) -> 01 1148.39/293.48 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 1148.39/293.48 a__sel1(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1148.39/293.48 a__quote(0) -> 01 1148.39/293.48 a__quote(s(X)) -> s1(a__quote(mark(X))) 1148.39/293.48 a__quote(dbl(X)) -> a__dbl1(mark(X)) 1148.39/293.48 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 1148.39/293.48 mark(dbl(X)) -> a__dbl(mark(X)) 1148.39/293.48 mark(dbls(X)) -> a__dbls(mark(X)) 1148.39/293.48 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1148.39/293.48 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 1148.39/293.48 mark(from(X)) -> a__from(X) 1148.39/293.48 mark(dbl1(X)) -> a__dbl1(mark(X)) 1148.39/293.48 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1148.39/293.48 mark(quote(X)) -> a__quote(mark(X)) 1148.39/293.48 mark(0) -> 0 1148.39/293.48 mark(s(X)) -> s(X) 1148.39/293.48 mark(nil) -> nil 1148.39/293.48 mark(cons(X1, X2)) -> cons(X1, X2) 1148.39/293.48 mark(01) -> 01 1148.39/293.48 mark(s1(X)) -> s1(mark(X)) 1148.39/293.48 a__dbl(X) -> dbl(X) 1148.39/293.48 a__dbls(X) -> dbls(X) 1148.39/293.48 a__sel(X1, X2) -> sel(X1, X2) 1148.39/293.48 a__indx(X1, X2) -> indx(X1, X2) 1148.39/293.48 a__from(X) -> from(X) 1148.39/293.48 a__dbl1(X) -> dbl1(X) 1148.39/293.48 a__sel1(X1, X2) -> sel1(X1, X2) 1148.39/293.48 a__quote(X) -> quote(X) 1148.39/293.48 1148.39/293.48 S is empty. 1148.39/293.48 Rewrite Strategy: INNERMOST 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1148.39/293.48 Transformed a relative TRS into a decreasing-loop problem. 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (2) 1148.39/293.48 Obligation: 1148.39/293.48 Analyzing the following TRS for decreasing loops: 1148.39/293.48 1148.39/293.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1148.39/293.48 1148.39/293.48 1148.39/293.48 The TRS R consists of the following rules: 1148.39/293.48 1148.39/293.48 a__dbl(0) -> 0 1148.39/293.48 a__dbl(s(X)) -> s(s(dbl(X))) 1148.39/293.48 a__dbls(nil) -> nil 1148.39/293.48 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 1148.39/293.48 a__sel(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1148.39/293.48 a__indx(nil, X) -> nil 1148.39/293.48 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 1148.39/293.48 a__from(X) -> cons(X, from(s(X))) 1148.39/293.48 a__dbl1(0) -> 01 1148.39/293.48 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 1148.39/293.48 a__sel1(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1148.39/293.48 a__quote(0) -> 01 1148.39/293.48 a__quote(s(X)) -> s1(a__quote(mark(X))) 1148.39/293.48 a__quote(dbl(X)) -> a__dbl1(mark(X)) 1148.39/293.48 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 1148.39/293.48 mark(dbl(X)) -> a__dbl(mark(X)) 1148.39/293.48 mark(dbls(X)) -> a__dbls(mark(X)) 1148.39/293.48 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1148.39/293.48 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 1148.39/293.48 mark(from(X)) -> a__from(X) 1148.39/293.48 mark(dbl1(X)) -> a__dbl1(mark(X)) 1148.39/293.48 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1148.39/293.48 mark(quote(X)) -> a__quote(mark(X)) 1148.39/293.48 mark(0) -> 0 1148.39/293.48 mark(s(X)) -> s(X) 1148.39/293.48 mark(nil) -> nil 1148.39/293.48 mark(cons(X1, X2)) -> cons(X1, X2) 1148.39/293.48 mark(01) -> 01 1148.39/293.48 mark(s1(X)) -> s1(mark(X)) 1148.39/293.48 a__dbl(X) -> dbl(X) 1148.39/293.48 a__dbls(X) -> dbls(X) 1148.39/293.48 a__sel(X1, X2) -> sel(X1, X2) 1148.39/293.48 a__indx(X1, X2) -> indx(X1, X2) 1148.39/293.48 a__from(X) -> from(X) 1148.39/293.48 a__dbl1(X) -> dbl1(X) 1148.39/293.48 a__sel1(X1, X2) -> sel1(X1, X2) 1148.39/293.48 a__quote(X) -> quote(X) 1148.39/293.48 1148.39/293.48 S is empty. 1148.39/293.48 Rewrite Strategy: INNERMOST 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1148.39/293.48 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1148.39/293.48 1148.39/293.48 The rewrite sequence 1148.39/293.48 1148.39/293.48 mark(indx(X1, X2)) ->^+ a__indx(mark(X1), X2) 1148.39/293.48 1148.39/293.48 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1148.39/293.48 1148.39/293.48 The pumping substitution is [X1 / indx(X1, X2)]. 1148.39/293.48 1148.39/293.48 The result substitution is [ ]. 1148.39/293.48 1148.39/293.48 1148.39/293.48 1148.39/293.48 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (4) 1148.39/293.48 Complex Obligation (BEST) 1148.39/293.48 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (5) 1148.39/293.48 Obligation: 1148.39/293.48 Proved the lower bound n^1 for the following obligation: 1148.39/293.48 1148.39/293.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1148.39/293.48 1148.39/293.48 1148.39/293.48 The TRS R consists of the following rules: 1148.39/293.48 1148.39/293.48 a__dbl(0) -> 0 1148.39/293.48 a__dbl(s(X)) -> s(s(dbl(X))) 1148.39/293.48 a__dbls(nil) -> nil 1148.39/293.48 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 1148.39/293.48 a__sel(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1148.39/293.48 a__indx(nil, X) -> nil 1148.39/293.48 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 1148.39/293.48 a__from(X) -> cons(X, from(s(X))) 1148.39/293.48 a__dbl1(0) -> 01 1148.39/293.48 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 1148.39/293.48 a__sel1(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1148.39/293.48 a__quote(0) -> 01 1148.39/293.48 a__quote(s(X)) -> s1(a__quote(mark(X))) 1148.39/293.48 a__quote(dbl(X)) -> a__dbl1(mark(X)) 1148.39/293.48 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 1148.39/293.48 mark(dbl(X)) -> a__dbl(mark(X)) 1148.39/293.48 mark(dbls(X)) -> a__dbls(mark(X)) 1148.39/293.48 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1148.39/293.48 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 1148.39/293.48 mark(from(X)) -> a__from(X) 1148.39/293.48 mark(dbl1(X)) -> a__dbl1(mark(X)) 1148.39/293.48 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1148.39/293.48 mark(quote(X)) -> a__quote(mark(X)) 1148.39/293.48 mark(0) -> 0 1148.39/293.48 mark(s(X)) -> s(X) 1148.39/293.48 mark(nil) -> nil 1148.39/293.48 mark(cons(X1, X2)) -> cons(X1, X2) 1148.39/293.48 mark(01) -> 01 1148.39/293.48 mark(s1(X)) -> s1(mark(X)) 1148.39/293.48 a__dbl(X) -> dbl(X) 1148.39/293.48 a__dbls(X) -> dbls(X) 1148.39/293.48 a__sel(X1, X2) -> sel(X1, X2) 1148.39/293.48 a__indx(X1, X2) -> indx(X1, X2) 1148.39/293.48 a__from(X) -> from(X) 1148.39/293.48 a__dbl1(X) -> dbl1(X) 1148.39/293.48 a__sel1(X1, X2) -> sel1(X1, X2) 1148.39/293.48 a__quote(X) -> quote(X) 1148.39/293.48 1148.39/293.48 S is empty. 1148.39/293.48 Rewrite Strategy: INNERMOST 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (6) LowerBoundPropagationProof (FINISHED) 1148.39/293.48 Propagated lower bound. 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (7) 1148.39/293.48 BOUNDS(n^1, INF) 1148.39/293.48 1148.39/293.48 ---------------------------------------- 1148.39/293.48 1148.39/293.48 (8) 1148.39/293.48 Obligation: 1148.39/293.48 Analyzing the following TRS for decreasing loops: 1148.39/293.48 1148.39/293.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1148.39/293.48 1148.39/293.48 1148.39/293.48 The TRS R consists of the following rules: 1148.39/293.48 1148.39/293.48 a__dbl(0) -> 0 1148.39/293.48 a__dbl(s(X)) -> s(s(dbl(X))) 1148.39/293.48 a__dbls(nil) -> nil 1148.39/293.48 a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) 1148.39/293.48 a__sel(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1148.39/293.48 a__indx(nil, X) -> nil 1148.39/293.48 a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) 1148.39/293.48 a__from(X) -> cons(X, from(s(X))) 1148.39/293.48 a__dbl1(0) -> 01 1148.39/293.48 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) 1148.39/293.48 a__sel1(0, cons(X, Y)) -> mark(X) 1148.39/293.48 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1148.39/293.48 a__quote(0) -> 01 1148.39/293.48 a__quote(s(X)) -> s1(a__quote(mark(X))) 1148.39/293.48 a__quote(dbl(X)) -> a__dbl1(mark(X)) 1148.39/293.48 a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) 1148.39/293.48 mark(dbl(X)) -> a__dbl(mark(X)) 1148.39/293.48 mark(dbls(X)) -> a__dbls(mark(X)) 1148.39/293.48 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1148.39/293.48 mark(indx(X1, X2)) -> a__indx(mark(X1), X2) 1148.39/293.48 mark(from(X)) -> a__from(X) 1148.39/293.48 mark(dbl1(X)) -> a__dbl1(mark(X)) 1148.39/293.48 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1148.39/293.48 mark(quote(X)) -> a__quote(mark(X)) 1148.39/293.48 mark(0) -> 0 1148.39/293.48 mark(s(X)) -> s(X) 1148.39/293.48 mark(nil) -> nil 1148.39/293.48 mark(cons(X1, X2)) -> cons(X1, X2) 1148.39/293.48 mark(01) -> 01 1148.39/293.48 mark(s1(X)) -> s1(mark(X)) 1148.39/293.48 a__dbl(X) -> dbl(X) 1148.39/293.48 a__dbls(X) -> dbls(X) 1148.39/293.48 a__sel(X1, X2) -> sel(X1, X2) 1148.39/293.48 a__indx(X1, X2) -> indx(X1, X2) 1148.39/293.48 a__from(X) -> from(X) 1148.39/293.48 a__dbl1(X) -> dbl1(X) 1148.39/293.48 a__sel1(X1, X2) -> sel1(X1, X2) 1148.39/293.48 a__quote(X) -> quote(X) 1148.39/293.48 1148.39/293.48 S is empty. 1148.39/293.48 Rewrite Strategy: INNERMOST 1148.49/293.55 EOF