3.42/1.62 WORST_CASE(Omega(n^1), O(n^1)) 3.50/1.63 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 3.50/1.63 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.50/1.63 3.50/1.63 3.50/1.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.50/1.63 3.50/1.63 (0) CpxTRS 3.50/1.63 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 3.50/1.63 (2) CpxTRS 3.50/1.63 (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms] 3.50/1.63 (4) BOUNDS(1, n^1) 3.50/1.63 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 3.50/1.63 (6) TRS for Loop Detection 3.50/1.63 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 3.50/1.63 (8) BEST 3.50/1.63 (9) proven lower bound 3.50/1.63 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 3.50/1.63 (11) BOUNDS(n^1, INF) 3.50/1.63 (12) TRS for Loop Detection 3.50/1.63 3.50/1.63 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (0) 3.50/1.63 Obligation: 3.50/1.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.50/1.63 3.50/1.63 3.50/1.63 The TRS R consists of the following rules: 3.50/1.63 3.50/1.63 f(a, empty) -> g(a, empty) 3.50/1.63 f(a, cons(x, k)) -> f(cons(x, a), k) 3.50/1.63 g(empty, d) -> d 3.50/1.63 g(cons(x, k), d) -> g(k, cons(x, d)) 3.50/1.63 3.50/1.63 S is empty. 3.50/1.63 Rewrite Strategy: INNERMOST 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 3.50/1.63 transformed relative TRS to TRS 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (2) 3.50/1.63 Obligation: 3.50/1.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 3.50/1.63 3.50/1.63 3.50/1.63 The TRS R consists of the following rules: 3.50/1.63 3.50/1.63 f(a, empty) -> g(a, empty) 3.50/1.63 f(a, cons(x, k)) -> f(cons(x, a), k) 3.50/1.63 g(empty, d) -> d 3.50/1.63 g(cons(x, k), d) -> g(k, cons(x, d)) 3.50/1.63 3.50/1.63 S is empty. 3.50/1.63 Rewrite Strategy: INNERMOST 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (3) CpxTrsMatchBoundsTAProof (FINISHED) 3.50/1.63 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 3.50/1.63 3.50/1.63 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 3.50/1.63 final states : [1, 2] 3.50/1.63 transitions: 3.50/1.63 empty0() -> 0 3.50/1.63 cons0(0, 0) -> 0 3.50/1.63 f0(0, 0) -> 1 3.50/1.63 g0(0, 0) -> 2 3.50/1.63 empty1() -> 3 3.50/1.63 g1(0, 3) -> 1 3.50/1.63 cons1(0, 0) -> 4 3.50/1.63 f1(4, 0) -> 1 3.50/1.63 cons1(0, 0) -> 5 3.50/1.63 g1(0, 5) -> 2 3.50/1.63 g1(4, 3) -> 1 3.50/1.63 cons1(0, 4) -> 4 3.50/1.63 cons1(0, 3) -> 5 3.50/1.63 g1(0, 5) -> 1 3.50/1.63 cons1(0, 5) -> 5 3.50/1.63 cons2(0, 3) -> 6 3.50/1.63 g2(0, 6) -> 1 3.50/1.63 g2(4, 6) -> 1 3.50/1.63 cons1(0, 6) -> 5 3.50/1.63 cons2(0, 6) -> 6 3.50/1.63 0 -> 2 3.50/1.63 3 -> 1 3.50/1.63 5 -> 2 3.50/1.63 5 -> 1 3.50/1.63 6 -> 1 3.50/1.63 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (4) 3.50/1.63 BOUNDS(1, n^1) 3.50/1.63 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 3.50/1.63 Transformed a relative TRS into a decreasing-loop problem. 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (6) 3.50/1.63 Obligation: 3.50/1.63 Analyzing the following TRS for decreasing loops: 3.50/1.63 3.50/1.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.50/1.63 3.50/1.63 3.50/1.63 The TRS R consists of the following rules: 3.50/1.63 3.50/1.63 f(a, empty) -> g(a, empty) 3.50/1.63 f(a, cons(x, k)) -> f(cons(x, a), k) 3.50/1.63 g(empty, d) -> d 3.50/1.63 g(cons(x, k), d) -> g(k, cons(x, d)) 3.50/1.63 3.50/1.63 S is empty. 3.50/1.63 Rewrite Strategy: INNERMOST 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (7) DecreasingLoopProof (LOWER BOUND(ID)) 3.50/1.63 The following loop(s) give(s) rise to the lower bound Omega(n^1): 3.50/1.63 3.50/1.63 The rewrite sequence 3.50/1.63 3.50/1.63 f(a, cons(x, k)) ->^+ f(cons(x, a), k) 3.50/1.63 3.50/1.63 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 3.50/1.63 3.50/1.63 The pumping substitution is [k / cons(x, k)]. 3.50/1.63 3.50/1.63 The result substitution is [a / cons(x, a)]. 3.50/1.63 3.50/1.63 3.50/1.63 3.50/1.63 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (8) 3.50/1.63 Complex Obligation (BEST) 3.50/1.63 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (9) 3.50/1.63 Obligation: 3.50/1.63 Proved the lower bound n^1 for the following obligation: 3.50/1.63 3.50/1.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.50/1.63 3.50/1.63 3.50/1.63 The TRS R consists of the following rules: 3.50/1.63 3.50/1.63 f(a, empty) -> g(a, empty) 3.50/1.63 f(a, cons(x, k)) -> f(cons(x, a), k) 3.50/1.63 g(empty, d) -> d 3.50/1.63 g(cons(x, k), d) -> g(k, cons(x, d)) 3.50/1.63 3.50/1.63 S is empty. 3.50/1.63 Rewrite Strategy: INNERMOST 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (10) LowerBoundPropagationProof (FINISHED) 3.50/1.63 Propagated lower bound. 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (11) 3.50/1.63 BOUNDS(n^1, INF) 3.50/1.63 3.50/1.63 ---------------------------------------- 3.50/1.63 3.50/1.63 (12) 3.50/1.63 Obligation: 3.50/1.63 Analyzing the following TRS for decreasing loops: 3.50/1.63 3.50/1.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.50/1.63 3.50/1.63 3.50/1.63 The TRS R consists of the following rules: 3.50/1.63 3.50/1.63 f(a, empty) -> g(a, empty) 3.50/1.63 f(a, cons(x, k)) -> f(cons(x, a), k) 3.50/1.63 g(empty, d) -> d 3.50/1.63 g(cons(x, k), d) -> g(k, cons(x, d)) 3.50/1.63 3.50/1.63 S is empty. 3.50/1.63 Rewrite Strategy: INNERMOST 3.50/1.67 EOF