22.73/7.06 WORST_CASE(Omega(n^1), O(n^1)) 22.73/7.07 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 22.73/7.07 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 22.73/7.07 22.73/7.07 22.73/7.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.73/7.07 22.73/7.07 (0) CpxTRS 22.73/7.07 (1) DependencyGraphProof [UPPER BOUND(ID), 0 ms] 22.73/7.07 (2) CpxTRS 22.73/7.07 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 22.73/7.07 (4) CpxTRS 22.73/7.07 (5) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms] 22.73/7.07 (6) BOUNDS(1, n^1) 22.73/7.07 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 22.73/7.07 (8) TRS for Loop Detection 22.73/7.07 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 22.73/7.07 (10) BEST 22.73/7.07 (11) proven lower bound 22.73/7.07 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 22.73/7.07 (13) BOUNDS(n^1, INF) 22.73/7.07 (14) TRS for Loop Detection 22.73/7.07 22.73/7.07 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (0) 22.73/7.07 Obligation: 22.73/7.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.73/7.07 22.73/7.07 22.73/7.07 The TRS R consists of the following rules: 22.73/7.07 22.73/7.07 g(c(x, s(y))) -> g(c(s(x), y)) 22.73/7.07 f(c(s(x), y)) -> f(c(x, s(y))) 22.73/7.07 f(f(x)) -> f(d(f(x))) 22.73/7.07 f(x) -> x 22.73/7.07 22.73/7.07 S is empty. 22.73/7.07 Rewrite Strategy: FULL 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (1) DependencyGraphProof (UPPER BOUND(ID)) 22.73/7.07 The following rules are not reachable from basic terms in the dependency graph and can be removed: 22.73/7.07 22.73/7.07 f(f(x)) -> f(d(f(x))) 22.73/7.07 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (2) 22.73/7.07 Obligation: 22.73/7.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 22.73/7.07 22.73/7.07 22.73/7.07 The TRS R consists of the following rules: 22.73/7.07 22.73/7.07 g(c(x, s(y))) -> g(c(s(x), y)) 22.73/7.07 f(c(s(x), y)) -> f(c(x, s(y))) 22.73/7.07 f(x) -> x 22.73/7.07 22.73/7.07 S is empty. 22.73/7.07 Rewrite Strategy: FULL 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 22.73/7.07 transformed relative TRS to TRS 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (4) 22.73/7.07 Obligation: 22.73/7.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 22.73/7.07 22.73/7.07 22.73/7.07 The TRS R consists of the following rules: 22.73/7.07 22.73/7.07 g(c(x, s(y))) -> g(c(s(x), y)) 22.73/7.07 f(c(s(x), y)) -> f(c(x, s(y))) 22.73/7.07 f(x) -> x 22.73/7.07 22.73/7.07 S is empty. 22.73/7.07 Rewrite Strategy: FULL 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (5) CpxTrsMatchBoundsTAProof (FINISHED) 22.73/7.07 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 1. 22.73/7.07 22.73/7.07 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 22.73/7.07 final states : [1, 2] 22.73/7.07 transitions: 22.73/7.07 c0(0, 0) -> 0 22.73/7.07 s0(0) -> 0 22.73/7.07 g0(0) -> 1 22.73/7.07 f0(0) -> 2 22.73/7.07 s1(0) -> 4 22.73/7.07 c1(4, 0) -> 3 22.73/7.07 g1(3) -> 1 22.73/7.07 s1(0) -> 6 22.73/7.07 c1(0, 6) -> 5 22.73/7.07 f1(5) -> 2 22.73/7.07 s1(4) -> 4 22.73/7.07 s1(6) -> 6 22.73/7.07 0 -> 2 22.73/7.07 5 -> 2 22.73/7.07 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (6) 22.73/7.07 BOUNDS(1, n^1) 22.73/7.07 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 22.73/7.07 Transformed a relative TRS into a decreasing-loop problem. 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (8) 22.73/7.07 Obligation: 22.73/7.07 Analyzing the following TRS for decreasing loops: 22.73/7.07 22.73/7.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.73/7.07 22.73/7.07 22.73/7.07 The TRS R consists of the following rules: 22.73/7.07 22.73/7.07 g(c(x, s(y))) -> g(c(s(x), y)) 22.73/7.07 f(c(s(x), y)) -> f(c(x, s(y))) 22.73/7.07 f(f(x)) -> f(d(f(x))) 22.73/7.07 f(x) -> x 22.73/7.07 22.73/7.07 S is empty. 22.73/7.07 Rewrite Strategy: FULL 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (9) DecreasingLoopProof (LOWER BOUND(ID)) 22.73/7.07 The following loop(s) give(s) rise to the lower bound Omega(n^1): 22.73/7.07 22.73/7.07 The rewrite sequence 22.73/7.07 22.73/7.07 g(c(x, s(y))) ->^+ g(c(s(x), y)) 22.73/7.07 22.73/7.07 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 22.73/7.07 22.73/7.07 The pumping substitution is [y / s(y)]. 22.73/7.07 22.73/7.07 The result substitution is [x / s(x)]. 22.73/7.07 22.73/7.07 22.73/7.07 22.73/7.07 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (10) 22.73/7.07 Complex Obligation (BEST) 22.73/7.07 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (11) 22.73/7.07 Obligation: 22.73/7.07 Proved the lower bound n^1 for the following obligation: 22.73/7.07 22.73/7.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.73/7.07 22.73/7.07 22.73/7.07 The TRS R consists of the following rules: 22.73/7.07 22.73/7.07 g(c(x, s(y))) -> g(c(s(x), y)) 22.73/7.07 f(c(s(x), y)) -> f(c(x, s(y))) 22.73/7.07 f(f(x)) -> f(d(f(x))) 22.73/7.07 f(x) -> x 22.73/7.07 22.73/7.07 S is empty. 22.73/7.07 Rewrite Strategy: FULL 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (12) LowerBoundPropagationProof (FINISHED) 22.73/7.07 Propagated lower bound. 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (13) 22.73/7.07 BOUNDS(n^1, INF) 22.73/7.07 22.73/7.07 ---------------------------------------- 22.73/7.07 22.73/7.07 (14) 22.73/7.07 Obligation: 22.73/7.07 Analyzing the following TRS for decreasing loops: 22.73/7.07 22.73/7.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.73/7.07 22.73/7.07 22.73/7.07 The TRS R consists of the following rules: 22.73/7.07 22.73/7.07 g(c(x, s(y))) -> g(c(s(x), y)) 22.73/7.07 f(c(s(x), y)) -> f(c(x, s(y))) 22.73/7.07 f(f(x)) -> f(d(f(x))) 22.73/7.07 f(x) -> x 22.73/7.07 22.73/7.07 S is empty. 22.73/7.07 Rewrite Strategy: FULL 22.81/7.11 EOF