313.21/291.50 WORST_CASE(Omega(n^1), ?) 313.21/291.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 313.21/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 313.21/291.51 313.21/291.51 313.21/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.21/291.51 313.21/291.51 (0) CpxTRS 313.21/291.51 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 313.21/291.51 (2) TRS for Loop Detection 313.21/291.51 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 313.21/291.51 (4) BEST 313.21/291.51 (5) proven lower bound 313.21/291.51 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 313.21/291.51 (7) BOUNDS(n^1, INF) 313.21/291.51 (8) TRS for Loop Detection 313.21/291.51 313.21/291.51 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (0) 313.21/291.51 Obligation: 313.21/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.21/291.51 313.21/291.51 313.21/291.51 The TRS R consists of the following rules: 313.21/291.51 313.21/291.51 isLeaf(leaf) -> true 313.21/291.51 isLeaf(cons(u, v)) -> false 313.21/291.51 left(cons(u, v)) -> u 313.21/291.51 right(cons(u, v)) -> v 313.21/291.51 concat(leaf, y) -> y 313.21/291.51 concat(cons(u, v), y) -> cons(u, concat(v, y)) 313.21/291.51 less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) 313.21/291.51 if1(b, true, u, v) -> false 313.21/291.51 if1(b, false, u, v) -> if2(b, u, v) 313.21/291.51 if2(true, u, v) -> true 313.21/291.51 if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) 313.21/291.51 313.21/291.51 S is empty. 313.21/291.51 Rewrite Strategy: FULL 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 313.21/291.51 Transformed a relative TRS into a decreasing-loop problem. 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (2) 313.21/291.51 Obligation: 313.21/291.51 Analyzing the following TRS for decreasing loops: 313.21/291.51 313.21/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.21/291.51 313.21/291.51 313.21/291.51 The TRS R consists of the following rules: 313.21/291.51 313.21/291.51 isLeaf(leaf) -> true 313.21/291.51 isLeaf(cons(u, v)) -> false 313.21/291.51 left(cons(u, v)) -> u 313.21/291.51 right(cons(u, v)) -> v 313.21/291.51 concat(leaf, y) -> y 313.21/291.51 concat(cons(u, v), y) -> cons(u, concat(v, y)) 313.21/291.51 less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) 313.21/291.51 if1(b, true, u, v) -> false 313.21/291.51 if1(b, false, u, v) -> if2(b, u, v) 313.21/291.51 if2(true, u, v) -> true 313.21/291.51 if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) 313.21/291.51 313.21/291.51 S is empty. 313.21/291.51 Rewrite Strategy: FULL 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (3) DecreasingLoopProof (LOWER BOUND(ID)) 313.21/291.51 The following loop(s) give(s) rise to the lower bound Omega(n^1): 313.21/291.51 313.21/291.51 The rewrite sequence 313.21/291.51 313.21/291.51 concat(cons(u, v), y) ->^+ cons(u, concat(v, y)) 313.21/291.51 313.21/291.51 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 313.21/291.51 313.21/291.51 The pumping substitution is [v / cons(u, v)]. 313.21/291.51 313.21/291.51 The result substitution is [ ]. 313.21/291.51 313.21/291.51 313.21/291.51 313.21/291.51 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (4) 313.21/291.51 Complex Obligation (BEST) 313.21/291.51 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (5) 313.21/291.51 Obligation: 313.21/291.51 Proved the lower bound n^1 for the following obligation: 313.21/291.51 313.21/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.21/291.51 313.21/291.51 313.21/291.51 The TRS R consists of the following rules: 313.21/291.51 313.21/291.51 isLeaf(leaf) -> true 313.21/291.51 isLeaf(cons(u, v)) -> false 313.21/291.51 left(cons(u, v)) -> u 313.21/291.51 right(cons(u, v)) -> v 313.21/291.51 concat(leaf, y) -> y 313.21/291.51 concat(cons(u, v), y) -> cons(u, concat(v, y)) 313.21/291.51 less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) 313.21/291.51 if1(b, true, u, v) -> false 313.21/291.51 if1(b, false, u, v) -> if2(b, u, v) 313.21/291.51 if2(true, u, v) -> true 313.21/291.51 if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) 313.21/291.51 313.21/291.51 S is empty. 313.21/291.51 Rewrite Strategy: FULL 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (6) LowerBoundPropagationProof (FINISHED) 313.21/291.51 Propagated lower bound. 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (7) 313.21/291.51 BOUNDS(n^1, INF) 313.21/291.51 313.21/291.51 ---------------------------------------- 313.21/291.51 313.21/291.51 (8) 313.21/291.51 Obligation: 313.21/291.51 Analyzing the following TRS for decreasing loops: 313.21/291.51 313.21/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.21/291.51 313.21/291.51 313.21/291.51 The TRS R consists of the following rules: 313.21/291.51 313.21/291.51 isLeaf(leaf) -> true 313.21/291.51 isLeaf(cons(u, v)) -> false 313.21/291.51 left(cons(u, v)) -> u 313.21/291.51 right(cons(u, v)) -> v 313.21/291.51 concat(leaf, y) -> y 313.21/291.51 concat(cons(u, v), y) -> cons(u, concat(v, y)) 313.21/291.51 less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) 313.21/291.51 if1(b, true, u, v) -> false 313.21/291.51 if1(b, false, u, v) -> if2(b, u, v) 313.21/291.51 if2(true, u, v) -> true 313.21/291.51 if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) 313.21/291.51 313.21/291.51 S is empty. 313.21/291.51 Rewrite Strategy: FULL 313.21/291.53 EOF