306.46/291.58 WORST_CASE(Omega(n^1), ?) 306.46/291.58 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 306.46/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 306.46/291.58 306.46/291.58 306.46/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.46/291.58 306.46/291.58 (0) CpxTRS 306.46/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 306.46/291.58 (2) TRS for Loop Detection 306.46/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 306.46/291.58 (4) BEST 306.46/291.58 (5) proven lower bound 306.46/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 306.46/291.58 (7) BOUNDS(n^1, INF) 306.46/291.58 (8) TRS for Loop Detection 306.46/291.58 306.46/291.58 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (0) 306.46/291.58 Obligation: 306.46/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.46/291.58 306.46/291.58 306.46/291.58 The TRS R consists of the following rules: 306.46/291.58 306.46/291.58 cond1(true, x, y) -> cond2(gr(y, 0), x, y) 306.46/291.58 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) 306.46/291.58 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) 306.46/291.58 gr(0, x) -> false 306.46/291.58 gr(s(x), 0) -> true 306.46/291.58 gr(s(x), s(y)) -> gr(x, y) 306.46/291.58 p(0) -> 0 306.46/291.58 p(s(x)) -> x 306.46/291.58 eq(0, 0) -> true 306.46/291.58 eq(s(x), 0) -> false 306.46/291.58 eq(0, s(x)) -> false 306.46/291.58 eq(s(x), s(y)) -> eq(x, y) 306.46/291.58 and(true, true) -> true 306.46/291.58 and(false, x) -> false 306.46/291.58 and(x, false) -> false 306.46/291.58 306.46/291.58 S is empty. 306.46/291.58 Rewrite Strategy: FULL 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 306.46/291.58 Transformed a relative TRS into a decreasing-loop problem. 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (2) 306.46/291.58 Obligation: 306.46/291.58 Analyzing the following TRS for decreasing loops: 306.46/291.58 306.46/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.46/291.58 306.46/291.58 306.46/291.58 The TRS R consists of the following rules: 306.46/291.58 306.46/291.58 cond1(true, x, y) -> cond2(gr(y, 0), x, y) 306.46/291.58 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) 306.46/291.58 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) 306.46/291.58 gr(0, x) -> false 306.46/291.58 gr(s(x), 0) -> true 306.46/291.58 gr(s(x), s(y)) -> gr(x, y) 306.46/291.58 p(0) -> 0 306.46/291.58 p(s(x)) -> x 306.46/291.58 eq(0, 0) -> true 306.46/291.58 eq(s(x), 0) -> false 306.46/291.58 eq(0, s(x)) -> false 306.46/291.58 eq(s(x), s(y)) -> eq(x, y) 306.46/291.58 and(true, true) -> true 306.46/291.58 and(false, x) -> false 306.46/291.58 and(x, false) -> false 306.46/291.58 306.46/291.58 S is empty. 306.46/291.58 Rewrite Strategy: FULL 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 306.46/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 306.46/291.58 306.46/291.58 The rewrite sequence 306.46/291.58 306.46/291.58 gr(s(x), s(y)) ->^+ gr(x, y) 306.46/291.58 306.46/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 306.46/291.58 306.46/291.58 The pumping substitution is [x / s(x), y / s(y)]. 306.46/291.58 306.46/291.58 The result substitution is [ ]. 306.46/291.58 306.46/291.58 306.46/291.58 306.46/291.58 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (4) 306.46/291.58 Complex Obligation (BEST) 306.46/291.58 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (5) 306.46/291.58 Obligation: 306.46/291.58 Proved the lower bound n^1 for the following obligation: 306.46/291.58 306.46/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.46/291.58 306.46/291.58 306.46/291.58 The TRS R consists of the following rules: 306.46/291.58 306.46/291.58 cond1(true, x, y) -> cond2(gr(y, 0), x, y) 306.46/291.58 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) 306.46/291.58 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) 306.46/291.58 gr(0, x) -> false 306.46/291.58 gr(s(x), 0) -> true 306.46/291.58 gr(s(x), s(y)) -> gr(x, y) 306.46/291.58 p(0) -> 0 306.46/291.58 p(s(x)) -> x 306.46/291.58 eq(0, 0) -> true 306.46/291.58 eq(s(x), 0) -> false 306.46/291.58 eq(0, s(x)) -> false 306.46/291.58 eq(s(x), s(y)) -> eq(x, y) 306.46/291.58 and(true, true) -> true 306.46/291.58 and(false, x) -> false 306.46/291.58 and(x, false) -> false 306.46/291.58 306.46/291.58 S is empty. 306.46/291.58 Rewrite Strategy: FULL 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (6) LowerBoundPropagationProof (FINISHED) 306.46/291.58 Propagated lower bound. 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (7) 306.46/291.58 BOUNDS(n^1, INF) 306.46/291.58 306.46/291.58 ---------------------------------------- 306.46/291.58 306.46/291.58 (8) 306.46/291.58 Obligation: 306.46/291.58 Analyzing the following TRS for decreasing loops: 306.46/291.58 306.46/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.46/291.58 306.46/291.58 306.46/291.58 The TRS R consists of the following rules: 306.46/291.58 306.46/291.58 cond1(true, x, y) -> cond2(gr(y, 0), x, y) 306.46/291.58 cond2(true, x, y) -> cond2(gr(y, 0), p(x), p(y)) 306.46/291.58 cond2(false, x, y) -> cond1(and(eq(x, y), gr(x, 0)), x, y) 306.46/291.58 gr(0, x) -> false 306.46/291.58 gr(s(x), 0) -> true 306.46/291.58 gr(s(x), s(y)) -> gr(x, y) 306.46/291.58 p(0) -> 0 306.46/291.58 p(s(x)) -> x 306.46/291.58 eq(0, 0) -> true 306.46/291.58 eq(s(x), 0) -> false 306.46/291.58 eq(0, s(x)) -> false 306.46/291.58 eq(s(x), s(y)) -> eq(x, y) 306.46/291.58 and(true, true) -> true 306.46/291.58 and(false, x) -> false 306.46/291.58 and(x, false) -> false 306.46/291.58 306.46/291.58 S is empty. 306.46/291.58 Rewrite Strategy: FULL 306.58/291.62 EOF