1121.47/291.56 WORST_CASE(Omega(n^1), ?) 1121.84/291.70 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1121.84/291.70 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1121.84/291.70 1121.84/291.70 1121.84/291.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1121.84/291.70 1121.84/291.70 (0) CpxTRS 1121.84/291.70 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1121.84/291.70 (2) TRS for Loop Detection 1121.84/291.70 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1121.84/291.70 (4) BEST 1121.84/291.70 (5) proven lower bound 1121.84/291.70 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1121.84/291.70 (7) BOUNDS(n^1, INF) 1121.84/291.70 (8) TRS for Loop Detection 1121.84/291.70 1121.84/291.70 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (0) 1121.84/291.70 Obligation: 1121.84/291.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1121.84/291.70 1121.84/291.70 1121.84/291.70 The TRS R consists of the following rules: 1121.84/291.70 1121.84/291.70 times(x, y) -> sum(generate(x, y)) 1121.84/291.70 generate(x, y) -> gen(x, y, 0) 1121.84/291.70 gen(x, y, z) -> if(ge(z, x), x, y, z) 1121.84/291.70 if(true, x, y, z) -> nil 1121.84/291.70 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1121.84/291.70 sum(xs) -> sum2(xs, 0) 1121.84/291.70 sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) 1121.84/291.70 ifsum(true, b, xs, y) -> y 1121.84/291.70 ifsum(false, b, xs, y) -> ifsum2(b, xs, y) 1121.84/291.70 ifsum2(true, xs, y) -> sum2(tail(xs), y) 1121.84/291.70 ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) 1121.84/291.70 isNil(nil) -> true 1121.84/291.70 isNil(cons(x, xs)) -> false 1121.84/291.70 tail(nil) -> nil 1121.84/291.70 tail(cons(x, xs)) -> xs 1121.84/291.70 head(cons(x, xs)) -> x 1121.84/291.70 head(nil) -> error 1121.84/291.70 isZero(0) -> true 1121.84/291.70 isZero(s(0)) -> false 1121.84/291.70 isZero(s(s(x))) -> isZero(s(x)) 1121.84/291.70 p(0) -> s(s(0)) 1121.84/291.70 p(s(0)) -> 0 1121.84/291.70 p(s(s(x))) -> s(p(s(x))) 1121.84/291.70 ge(x, 0) -> true 1121.84/291.70 ge(0, s(y)) -> false 1121.84/291.70 ge(s(x), s(y)) -> ge(x, y) 1121.84/291.70 a -> c 1121.84/291.70 a -> d 1121.84/291.70 1121.84/291.70 S is empty. 1121.84/291.70 Rewrite Strategy: INNERMOST 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1121.84/291.70 Transformed a relative TRS into a decreasing-loop problem. 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (2) 1121.84/291.70 Obligation: 1121.84/291.70 Analyzing the following TRS for decreasing loops: 1121.84/291.70 1121.84/291.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1121.84/291.70 1121.84/291.70 1121.84/291.70 The TRS R consists of the following rules: 1121.84/291.70 1121.84/291.70 times(x, y) -> sum(generate(x, y)) 1121.84/291.70 generate(x, y) -> gen(x, y, 0) 1121.84/291.70 gen(x, y, z) -> if(ge(z, x), x, y, z) 1121.84/291.70 if(true, x, y, z) -> nil 1121.84/291.70 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1121.84/291.70 sum(xs) -> sum2(xs, 0) 1121.84/291.70 sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) 1121.84/291.70 ifsum(true, b, xs, y) -> y 1121.84/291.70 ifsum(false, b, xs, y) -> ifsum2(b, xs, y) 1121.84/291.70 ifsum2(true, xs, y) -> sum2(tail(xs), y) 1121.84/291.70 ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) 1121.84/291.70 isNil(nil) -> true 1121.84/291.70 isNil(cons(x, xs)) -> false 1121.84/291.70 tail(nil) -> nil 1121.84/291.70 tail(cons(x, xs)) -> xs 1121.84/291.70 head(cons(x, xs)) -> x 1121.84/291.70 head(nil) -> error 1121.84/291.70 isZero(0) -> true 1121.84/291.70 isZero(s(0)) -> false 1121.84/291.70 isZero(s(s(x))) -> isZero(s(x)) 1121.84/291.70 p(0) -> s(s(0)) 1121.84/291.70 p(s(0)) -> 0 1121.84/291.70 p(s(s(x))) -> s(p(s(x))) 1121.84/291.70 ge(x, 0) -> true 1121.84/291.70 ge(0, s(y)) -> false 1121.84/291.70 ge(s(x), s(y)) -> ge(x, y) 1121.84/291.70 a -> c 1121.84/291.70 a -> d 1121.84/291.70 1121.84/291.70 S is empty. 1121.84/291.70 Rewrite Strategy: INNERMOST 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1121.84/291.70 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1121.84/291.70 1121.84/291.70 The rewrite sequence 1121.84/291.70 1121.84/291.70 p(s(s(x))) ->^+ s(p(s(x))) 1121.84/291.70 1121.84/291.70 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1121.84/291.70 1121.84/291.70 The pumping substitution is [x / s(x)]. 1121.84/291.70 1121.84/291.70 The result substitution is [ ]. 1121.84/291.70 1121.84/291.70 1121.84/291.70 1121.84/291.70 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (4) 1121.84/291.70 Complex Obligation (BEST) 1121.84/291.70 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (5) 1121.84/291.70 Obligation: 1121.84/291.70 Proved the lower bound n^1 for the following obligation: 1121.84/291.70 1121.84/291.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1121.84/291.70 1121.84/291.70 1121.84/291.70 The TRS R consists of the following rules: 1121.84/291.70 1121.84/291.70 times(x, y) -> sum(generate(x, y)) 1121.84/291.70 generate(x, y) -> gen(x, y, 0) 1121.84/291.70 gen(x, y, z) -> if(ge(z, x), x, y, z) 1121.84/291.70 if(true, x, y, z) -> nil 1121.84/291.70 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1121.84/291.70 sum(xs) -> sum2(xs, 0) 1121.84/291.70 sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) 1121.84/291.70 ifsum(true, b, xs, y) -> y 1121.84/291.70 ifsum(false, b, xs, y) -> ifsum2(b, xs, y) 1121.84/291.70 ifsum2(true, xs, y) -> sum2(tail(xs), y) 1121.84/291.70 ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) 1121.84/291.70 isNil(nil) -> true 1121.84/291.70 isNil(cons(x, xs)) -> false 1121.84/291.70 tail(nil) -> nil 1121.84/291.70 tail(cons(x, xs)) -> xs 1121.84/291.70 head(cons(x, xs)) -> x 1121.84/291.70 head(nil) -> error 1121.84/291.70 isZero(0) -> true 1121.84/291.70 isZero(s(0)) -> false 1121.84/291.70 isZero(s(s(x))) -> isZero(s(x)) 1121.84/291.70 p(0) -> s(s(0)) 1121.84/291.70 p(s(0)) -> 0 1121.84/291.70 p(s(s(x))) -> s(p(s(x))) 1121.84/291.70 ge(x, 0) -> true 1121.84/291.70 ge(0, s(y)) -> false 1121.84/291.70 ge(s(x), s(y)) -> ge(x, y) 1121.84/291.70 a -> c 1121.84/291.70 a -> d 1121.84/291.70 1121.84/291.70 S is empty. 1121.84/291.70 Rewrite Strategy: INNERMOST 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (6) LowerBoundPropagationProof (FINISHED) 1121.84/291.70 Propagated lower bound. 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (7) 1121.84/291.70 BOUNDS(n^1, INF) 1121.84/291.70 1121.84/291.70 ---------------------------------------- 1121.84/291.70 1121.84/291.70 (8) 1121.84/291.70 Obligation: 1121.84/291.70 Analyzing the following TRS for decreasing loops: 1121.84/291.70 1121.84/291.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1121.84/291.70 1121.84/291.70 1121.84/291.70 The TRS R consists of the following rules: 1121.84/291.70 1121.84/291.70 times(x, y) -> sum(generate(x, y)) 1121.84/291.70 generate(x, y) -> gen(x, y, 0) 1121.84/291.70 gen(x, y, z) -> if(ge(z, x), x, y, z) 1121.84/291.70 if(true, x, y, z) -> nil 1121.84/291.70 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1121.84/291.70 sum(xs) -> sum2(xs, 0) 1121.84/291.70 sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) 1121.84/291.70 ifsum(true, b, xs, y) -> y 1121.84/291.70 ifsum(false, b, xs, y) -> ifsum2(b, xs, y) 1121.84/291.70 ifsum2(true, xs, y) -> sum2(tail(xs), y) 1121.84/291.70 ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) 1121.84/291.70 isNil(nil) -> true 1121.84/291.70 isNil(cons(x, xs)) -> false 1121.84/291.70 tail(nil) -> nil 1121.84/291.70 tail(cons(x, xs)) -> xs 1121.84/291.70 head(cons(x, xs)) -> x 1121.84/291.70 head(nil) -> error 1121.84/291.70 isZero(0) -> true 1121.84/291.70 isZero(s(0)) -> false 1121.84/291.70 isZero(s(s(x))) -> isZero(s(x)) 1121.84/291.70 p(0) -> s(s(0)) 1121.84/291.70 p(s(0)) -> 0 1121.84/291.70 p(s(s(x))) -> s(p(s(x))) 1121.84/291.70 ge(x, 0) -> true 1121.84/291.70 ge(0, s(y)) -> false 1121.84/291.70 ge(s(x), s(y)) -> ge(x, y) 1121.84/291.70 a -> c 1121.84/291.70 a -> d 1121.84/291.70 1121.84/291.70 S is empty. 1121.84/291.70 Rewrite Strategy: INNERMOST 1122.07/291.78 EOF