305.68/291.46 WORST_CASE(Omega(n^1), ?) 305.68/291.47 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 305.68/291.47 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.68/291.47 305.68/291.47 305.68/291.47 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.68/291.47 305.68/291.47 (0) CpxTRS 305.68/291.47 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 305.68/291.47 (2) TRS for Loop Detection 305.68/291.47 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 305.68/291.47 (4) BEST 305.68/291.47 (5) proven lower bound 305.68/291.47 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 305.68/291.47 (7) BOUNDS(n^1, INF) 305.68/291.47 (8) TRS for Loop Detection 305.68/291.47 305.68/291.47 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (0) 305.68/291.47 Obligation: 305.68/291.47 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.68/291.47 305.68/291.47 305.68/291.47 The TRS R consists of the following rules: 305.68/291.47 305.68/291.47 double(0) -> 0 305.68/291.47 double(s(x)) -> s(s(double(x))) 305.68/291.47 del(x, nil) -> nil 305.68/291.47 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 305.68/291.47 if(true, x, y, xs) -> xs 305.68/291.47 if(false, x, y, xs) -> cons(y, del(x, xs)) 305.68/291.47 eq(0, 0) -> true 305.68/291.47 eq(0, s(y)) -> false 305.68/291.47 eq(s(x), 0) -> false 305.68/291.47 eq(s(x), s(y)) -> eq(x, y) 305.68/291.47 first(nil) -> 0 305.68/291.47 first(cons(x, xs)) -> x 305.68/291.47 doublelist(nil) -> nil 305.68/291.47 doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) 305.68/291.47 305.68/291.47 S is empty. 305.68/291.47 Rewrite Strategy: FULL 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 305.68/291.47 Transformed a relative TRS into a decreasing-loop problem. 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (2) 305.68/291.47 Obligation: 305.68/291.47 Analyzing the following TRS for decreasing loops: 305.68/291.47 305.68/291.47 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.68/291.47 305.68/291.47 305.68/291.47 The TRS R consists of the following rules: 305.68/291.47 305.68/291.47 double(0) -> 0 305.68/291.47 double(s(x)) -> s(s(double(x))) 305.68/291.47 del(x, nil) -> nil 305.68/291.47 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 305.68/291.47 if(true, x, y, xs) -> xs 305.68/291.47 if(false, x, y, xs) -> cons(y, del(x, xs)) 305.68/291.47 eq(0, 0) -> true 305.68/291.47 eq(0, s(y)) -> false 305.68/291.47 eq(s(x), 0) -> false 305.68/291.47 eq(s(x), s(y)) -> eq(x, y) 305.68/291.47 first(nil) -> 0 305.68/291.47 first(cons(x, xs)) -> x 305.68/291.47 doublelist(nil) -> nil 305.68/291.47 doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) 305.68/291.47 305.68/291.47 S is empty. 305.68/291.47 Rewrite Strategy: FULL 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (3) DecreasingLoopProof (LOWER BOUND(ID)) 305.68/291.47 The following loop(s) give(s) rise to the lower bound Omega(n^1): 305.68/291.47 305.68/291.47 The rewrite sequence 305.68/291.47 305.68/291.47 double(s(x)) ->^+ s(s(double(x))) 305.68/291.47 305.68/291.47 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 305.68/291.47 305.68/291.47 The pumping substitution is [x / s(x)]. 305.68/291.47 305.68/291.47 The result substitution is [ ]. 305.68/291.47 305.68/291.47 305.68/291.47 305.68/291.47 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (4) 305.68/291.47 Complex Obligation (BEST) 305.68/291.47 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (5) 305.68/291.47 Obligation: 305.68/291.47 Proved the lower bound n^1 for the following obligation: 305.68/291.47 305.68/291.47 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.68/291.47 305.68/291.47 305.68/291.47 The TRS R consists of the following rules: 305.68/291.47 305.68/291.47 double(0) -> 0 305.68/291.47 double(s(x)) -> s(s(double(x))) 305.68/291.47 del(x, nil) -> nil 305.68/291.47 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 305.68/291.47 if(true, x, y, xs) -> xs 305.68/291.47 if(false, x, y, xs) -> cons(y, del(x, xs)) 305.68/291.47 eq(0, 0) -> true 305.68/291.47 eq(0, s(y)) -> false 305.68/291.47 eq(s(x), 0) -> false 305.68/291.47 eq(s(x), s(y)) -> eq(x, y) 305.68/291.47 first(nil) -> 0 305.68/291.47 first(cons(x, xs)) -> x 305.68/291.47 doublelist(nil) -> nil 305.68/291.47 doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) 305.68/291.47 305.68/291.47 S is empty. 305.68/291.47 Rewrite Strategy: FULL 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (6) LowerBoundPropagationProof (FINISHED) 305.68/291.47 Propagated lower bound. 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (7) 305.68/291.47 BOUNDS(n^1, INF) 305.68/291.47 305.68/291.47 ---------------------------------------- 305.68/291.47 305.68/291.47 (8) 305.68/291.47 Obligation: 305.68/291.47 Analyzing the following TRS for decreasing loops: 305.68/291.47 305.68/291.47 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.68/291.47 305.68/291.47 305.68/291.47 The TRS R consists of the following rules: 305.68/291.47 305.68/291.47 double(0) -> 0 305.68/291.47 double(s(x)) -> s(s(double(x))) 305.68/291.47 del(x, nil) -> nil 305.68/291.47 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 305.68/291.47 if(true, x, y, xs) -> xs 305.68/291.47 if(false, x, y, xs) -> cons(y, del(x, xs)) 305.68/291.47 eq(0, 0) -> true 305.68/291.47 eq(0, s(y)) -> false 305.68/291.47 eq(s(x), 0) -> false 305.68/291.47 eq(s(x), s(y)) -> eq(x, y) 305.68/291.47 first(nil) -> 0 305.68/291.47 first(cons(x, xs)) -> x 305.68/291.47 doublelist(nil) -> nil 305.68/291.47 doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) 305.68/291.47 305.68/291.47 S is empty. 305.68/291.47 Rewrite Strategy: FULL 305.68/291.49 EOF