3.51/2.03 WORST_CASE(Omega(n^1), O(n^1)) 3.51/2.05 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.51/2.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.51/2.05 3.51/2.05 3.51/2.05 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.51/2.05 3.51/2.05 (0) CpxTRS 3.51/2.05 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 3.51/2.05 (2) CpxTRS 3.51/2.05 (3) CpxTrsMatchBoundsTAProof [FINISHED, 35 ms] 3.51/2.05 (4) BOUNDS(1, n^1) 3.51/2.05 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 3.51/2.05 (6) TRS for Loop Detection 3.51/2.05 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 3.51/2.05 (8) BEST 3.51/2.05 (9) proven lower bound 3.51/2.05 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 3.51/2.05 (11) BOUNDS(n^1, INF) 3.51/2.05 (12) TRS for Loop Detection 3.51/2.05 3.51/2.05 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (0) 3.51/2.05 Obligation: 3.51/2.05 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.51/2.05 3.51/2.05 3.51/2.05 The TRS R consists of the following rules: 3.51/2.05 3.51/2.05 dbl(S(0), S(0)) -> S(S(S(S(0)))) 3.51/2.05 save(S(x)) -> dbl(0, save(x)) 3.51/2.05 save(0) -> 0 3.51/2.05 dbl(0, y) -> y 3.51/2.05 3.51/2.05 S is empty. 3.51/2.05 Rewrite Strategy: INNERMOST 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 3.51/2.05 transformed relative TRS to TRS 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (2) 3.51/2.05 Obligation: 3.51/2.05 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 3.51/2.05 3.51/2.05 3.51/2.05 The TRS R consists of the following rules: 3.51/2.05 3.51/2.05 dbl(S(0), S(0)) -> S(S(S(S(0)))) 3.51/2.05 save(S(x)) -> dbl(0, save(x)) 3.51/2.05 save(0) -> 0 3.51/2.05 dbl(0, y) -> y 3.51/2.05 3.51/2.05 S is empty. 3.51/2.05 Rewrite Strategy: INNERMOST 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (3) CpxTrsMatchBoundsTAProof (FINISHED) 3.51/2.05 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. 3.51/2.05 3.51/2.05 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 3.51/2.05 final states : [1, 2] 3.51/2.05 transitions: 3.51/2.05 S0(0) -> 0 3.51/2.05 00() -> 0 3.51/2.05 dbl0(0, 0) -> 1 3.51/2.05 save0(0) -> 2 3.51/2.05 01() -> 5 3.51/2.05 S1(5) -> 4 3.51/2.05 S1(4) -> 3 3.51/2.05 S1(3) -> 3 3.51/2.05 S1(3) -> 1 3.51/2.05 01() -> 6 3.51/2.05 save1(0) -> 7 3.51/2.05 dbl1(6, 7) -> 2 3.51/2.05 01() -> 2 3.51/2.05 dbl1(6, 7) -> 7 3.51/2.05 01() -> 7 3.51/2.05 0 -> 1 3.51/2.05 7 -> 2 3.51/2.05 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (4) 3.51/2.05 BOUNDS(1, n^1) 3.51/2.05 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 3.51/2.05 Transformed a relative TRS into a decreasing-loop problem. 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (6) 3.51/2.05 Obligation: 3.51/2.05 Analyzing the following TRS for decreasing loops: 3.51/2.05 3.51/2.05 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.51/2.05 3.51/2.05 3.51/2.05 The TRS R consists of the following rules: 3.51/2.05 3.51/2.05 dbl(S(0), S(0)) -> S(S(S(S(0)))) 3.51/2.05 save(S(x)) -> dbl(0, save(x)) 3.51/2.05 save(0) -> 0 3.51/2.05 dbl(0, y) -> y 3.51/2.05 3.51/2.05 S is empty. 3.51/2.05 Rewrite Strategy: INNERMOST 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (7) DecreasingLoopProof (LOWER BOUND(ID)) 3.51/2.05 The following loop(s) give(s) rise to the lower bound Omega(n^1): 3.51/2.05 3.51/2.05 The rewrite sequence 3.51/2.05 3.51/2.05 save(S(x)) ->^+ dbl(0, save(x)) 3.51/2.05 3.51/2.05 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 3.51/2.05 3.51/2.05 The pumping substitution is [x / S(x)]. 3.51/2.05 3.51/2.05 The result substitution is [ ]. 3.51/2.05 3.51/2.05 3.51/2.05 3.51/2.05 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (8) 3.51/2.05 Complex Obligation (BEST) 3.51/2.05 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (9) 3.51/2.05 Obligation: 3.51/2.05 Proved the lower bound n^1 for the following obligation: 3.51/2.05 3.51/2.05 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.51/2.05 3.51/2.05 3.51/2.05 The TRS R consists of the following rules: 3.51/2.05 3.51/2.05 dbl(S(0), S(0)) -> S(S(S(S(0)))) 3.51/2.05 save(S(x)) -> dbl(0, save(x)) 3.51/2.05 save(0) -> 0 3.51/2.05 dbl(0, y) -> y 3.51/2.05 3.51/2.05 S is empty. 3.51/2.05 Rewrite Strategy: INNERMOST 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (10) LowerBoundPropagationProof (FINISHED) 3.51/2.05 Propagated lower bound. 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (11) 3.51/2.05 BOUNDS(n^1, INF) 3.51/2.05 3.51/2.05 ---------------------------------------- 3.51/2.05 3.51/2.05 (12) 3.51/2.05 Obligation: 3.51/2.05 Analyzing the following TRS for decreasing loops: 3.51/2.05 3.51/2.05 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.51/2.05 3.51/2.05 3.51/2.05 The TRS R consists of the following rules: 3.51/2.05 3.51/2.05 dbl(S(0), S(0)) -> S(S(S(S(0)))) 3.51/2.05 save(S(x)) -> dbl(0, save(x)) 3.51/2.05 save(0) -> 0 3.51/2.05 dbl(0, y) -> y 3.51/2.05 3.51/2.05 S is empty. 3.51/2.05 Rewrite Strategy: INNERMOST 3.51/2.07 EOF