306.98/291.55 WORST_CASE(Omega(n^1), ?) 306.98/291.55 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 306.98/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 306.98/291.55 306.98/291.55 306.98/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.98/291.55 306.98/291.55 (0) CpxTRS 306.98/291.55 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 306.98/291.55 (2) TRS for Loop Detection 306.98/291.55 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 306.98/291.55 (4) BEST 306.98/291.55 (5) proven lower bound 306.98/291.55 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 306.98/291.55 (7) BOUNDS(n^1, INF) 306.98/291.55 (8) TRS for Loop Detection 306.98/291.55 306.98/291.55 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (0) 306.98/291.55 Obligation: 306.98/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.98/291.55 306.98/291.55 306.98/291.55 The TRS R consists of the following rules: 306.98/291.55 306.98/291.55 eq(0, 0) -> true 306.98/291.55 eq(0, s(x)) -> false 306.98/291.55 eq(s(x), 0) -> false 306.98/291.55 eq(s(x), s(y)) -> eq(x, y) 306.98/291.55 le(0, y) -> true 306.98/291.55 le(s(x), 0) -> false 306.98/291.55 le(s(x), s(y)) -> le(x, y) 306.98/291.55 app(nil, y) -> y 306.98/291.55 app(add(n, x), y) -> add(n, app(x, y)) 306.98/291.55 min(nil) -> 0 306.98/291.55 min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) 306.98/291.55 minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) 306.98/291.55 minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) 306.98/291.55 if_min(true, x, y, m) -> m 306.98/291.55 if_min(false, x, y, m) -> minIter(x, y, m) 306.98/291.55 head(add(n, x)) -> n 306.98/291.55 tail(add(n, x)) -> x 306.98/291.55 tail(nil) -> nil 306.98/291.55 null(nil) -> true 306.98/291.55 null(add(n, x)) -> false 306.98/291.55 rm(n, nil) -> nil 306.98/291.55 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 306.98/291.55 if_rm(true, n, add(m, x)) -> rm(n, x) 306.98/291.55 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 306.98/291.55 minsort(nil, nil) -> nil 306.98/291.55 minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) 306.98/291.55 if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) 306.98/291.55 if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) 306.98/291.55 306.98/291.55 S is empty. 306.98/291.55 Rewrite Strategy: FULL 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 306.98/291.55 Transformed a relative TRS into a decreasing-loop problem. 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (2) 306.98/291.55 Obligation: 306.98/291.55 Analyzing the following TRS for decreasing loops: 306.98/291.55 306.98/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.98/291.55 306.98/291.55 306.98/291.55 The TRS R consists of the following rules: 306.98/291.55 306.98/291.55 eq(0, 0) -> true 306.98/291.55 eq(0, s(x)) -> false 306.98/291.55 eq(s(x), 0) -> false 306.98/291.55 eq(s(x), s(y)) -> eq(x, y) 306.98/291.55 le(0, y) -> true 306.98/291.55 le(s(x), 0) -> false 306.98/291.55 le(s(x), s(y)) -> le(x, y) 306.98/291.55 app(nil, y) -> y 306.98/291.55 app(add(n, x), y) -> add(n, app(x, y)) 306.98/291.55 min(nil) -> 0 306.98/291.55 min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) 306.98/291.55 minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) 306.98/291.55 minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) 306.98/291.55 if_min(true, x, y, m) -> m 306.98/291.55 if_min(false, x, y, m) -> minIter(x, y, m) 306.98/291.55 head(add(n, x)) -> n 306.98/291.55 tail(add(n, x)) -> x 306.98/291.55 tail(nil) -> nil 306.98/291.55 null(nil) -> true 306.98/291.55 null(add(n, x)) -> false 306.98/291.55 rm(n, nil) -> nil 306.98/291.55 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 306.98/291.55 if_rm(true, n, add(m, x)) -> rm(n, x) 306.98/291.55 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 306.98/291.55 minsort(nil, nil) -> nil 306.98/291.55 minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) 306.98/291.55 if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) 306.98/291.55 if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) 306.98/291.55 306.98/291.55 S is empty. 306.98/291.55 Rewrite Strategy: FULL 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (3) DecreasingLoopProof (LOWER BOUND(ID)) 306.98/291.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 306.98/291.55 306.98/291.55 The rewrite sequence 306.98/291.55 306.98/291.55 le(s(x), s(y)) ->^+ le(x, y) 306.98/291.55 306.98/291.55 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 306.98/291.55 306.98/291.55 The pumping substitution is [x / s(x), y / s(y)]. 306.98/291.55 306.98/291.55 The result substitution is [ ]. 306.98/291.55 306.98/291.55 306.98/291.55 306.98/291.55 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (4) 306.98/291.55 Complex Obligation (BEST) 306.98/291.55 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (5) 306.98/291.55 Obligation: 306.98/291.55 Proved the lower bound n^1 for the following obligation: 306.98/291.55 306.98/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.98/291.55 306.98/291.55 306.98/291.55 The TRS R consists of the following rules: 306.98/291.55 306.98/291.55 eq(0, 0) -> true 306.98/291.55 eq(0, s(x)) -> false 306.98/291.55 eq(s(x), 0) -> false 306.98/291.55 eq(s(x), s(y)) -> eq(x, y) 306.98/291.55 le(0, y) -> true 306.98/291.55 le(s(x), 0) -> false 306.98/291.55 le(s(x), s(y)) -> le(x, y) 306.98/291.55 app(nil, y) -> y 306.98/291.55 app(add(n, x), y) -> add(n, app(x, y)) 306.98/291.55 min(nil) -> 0 306.98/291.55 min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) 306.98/291.55 minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) 306.98/291.55 minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) 306.98/291.55 if_min(true, x, y, m) -> m 306.98/291.55 if_min(false, x, y, m) -> minIter(x, y, m) 306.98/291.55 head(add(n, x)) -> n 306.98/291.55 tail(add(n, x)) -> x 306.98/291.55 tail(nil) -> nil 306.98/291.55 null(nil) -> true 306.98/291.55 null(add(n, x)) -> false 306.98/291.55 rm(n, nil) -> nil 306.98/291.55 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 306.98/291.55 if_rm(true, n, add(m, x)) -> rm(n, x) 306.98/291.55 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 306.98/291.55 minsort(nil, nil) -> nil 306.98/291.55 minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) 306.98/291.55 if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) 306.98/291.55 if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) 306.98/291.55 306.98/291.55 S is empty. 306.98/291.55 Rewrite Strategy: FULL 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (6) LowerBoundPropagationProof (FINISHED) 306.98/291.55 Propagated lower bound. 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (7) 306.98/291.55 BOUNDS(n^1, INF) 306.98/291.55 306.98/291.55 ---------------------------------------- 306.98/291.55 306.98/291.55 (8) 306.98/291.55 Obligation: 306.98/291.55 Analyzing the following TRS for decreasing loops: 306.98/291.55 306.98/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.98/291.55 306.98/291.55 306.98/291.55 The TRS R consists of the following rules: 306.98/291.55 306.98/291.55 eq(0, 0) -> true 306.98/291.55 eq(0, s(x)) -> false 306.98/291.55 eq(s(x), 0) -> false 306.98/291.55 eq(s(x), s(y)) -> eq(x, y) 306.98/291.55 le(0, y) -> true 306.98/291.55 le(s(x), 0) -> false 306.98/291.55 le(s(x), s(y)) -> le(x, y) 306.98/291.55 app(nil, y) -> y 306.98/291.55 app(add(n, x), y) -> add(n, app(x, y)) 306.98/291.55 min(nil) -> 0 306.98/291.55 min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) 306.98/291.55 minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) 306.98/291.55 minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) 306.98/291.55 if_min(true, x, y, m) -> m 306.98/291.55 if_min(false, x, y, m) -> minIter(x, y, m) 306.98/291.55 head(add(n, x)) -> n 306.98/291.55 tail(add(n, x)) -> x 306.98/291.55 tail(nil) -> nil 306.98/291.55 null(nil) -> true 306.98/291.55 null(add(n, x)) -> false 306.98/291.55 rm(n, nil) -> nil 306.98/291.55 rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) 306.98/291.55 if_rm(true, n, add(m, x)) -> rm(n, x) 306.98/291.55 if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) 306.98/291.55 minsort(nil, nil) -> nil 306.98/291.55 minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) 306.98/291.55 if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) 306.98/291.55 if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) 306.98/291.55 306.98/291.55 S is empty. 306.98/291.55 Rewrite Strategy: FULL 306.98/291.58 EOF