1131.70/291.59 WORST_CASE(Omega(n^1), ?) 1131.70/291.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1131.70/291.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1131.70/291.60 1131.70/291.60 1131.70/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.70/291.60 1131.70/291.60 (0) CpxTRS 1131.70/291.60 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1131.70/291.60 (2) TRS for Loop Detection 1131.70/291.60 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1131.70/291.60 (4) BEST 1131.70/291.60 (5) proven lower bound 1131.70/291.60 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1131.70/291.60 (7) BOUNDS(n^1, INF) 1131.70/291.60 (8) TRS for Loop Detection 1131.70/291.60 1131.70/291.60 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (0) 1131.70/291.60 Obligation: 1131.70/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.70/291.60 1131.70/291.60 1131.70/291.60 The TRS R consists of the following rules: 1131.70/291.60 1131.70/291.60 active(nats) -> mark(adx(zeros)) 1131.70/291.60 active(zeros) -> mark(cons(0, zeros)) 1131.70/291.60 active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y))) 1131.70/291.60 active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y)))) 1131.70/291.60 active(hd(cons(X, Y))) -> mark(X) 1131.70/291.60 active(tl(cons(X, Y))) -> mark(Y) 1131.70/291.60 active(adx(X)) -> adx(active(X)) 1131.70/291.60 active(incr(X)) -> incr(active(X)) 1131.70/291.60 active(hd(X)) -> hd(active(X)) 1131.70/291.60 active(tl(X)) -> tl(active(X)) 1131.70/291.60 adx(mark(X)) -> mark(adx(X)) 1131.70/291.60 incr(mark(X)) -> mark(incr(X)) 1131.70/291.60 hd(mark(X)) -> mark(hd(X)) 1131.70/291.60 tl(mark(X)) -> mark(tl(X)) 1131.70/291.60 proper(nats) -> ok(nats) 1131.70/291.60 proper(adx(X)) -> adx(proper(X)) 1131.70/291.60 proper(zeros) -> ok(zeros) 1131.70/291.60 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1131.70/291.60 proper(0) -> ok(0) 1131.70/291.60 proper(incr(X)) -> incr(proper(X)) 1131.70/291.60 proper(s(X)) -> s(proper(X)) 1131.70/291.60 proper(hd(X)) -> hd(proper(X)) 1131.70/291.60 proper(tl(X)) -> tl(proper(X)) 1131.70/291.60 adx(ok(X)) -> ok(adx(X)) 1131.70/291.60 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1131.70/291.60 incr(ok(X)) -> ok(incr(X)) 1131.70/291.60 s(ok(X)) -> ok(s(X)) 1131.70/291.60 hd(ok(X)) -> ok(hd(X)) 1131.70/291.60 tl(ok(X)) -> ok(tl(X)) 1131.70/291.60 top(mark(X)) -> top(proper(X)) 1131.70/291.60 top(ok(X)) -> top(active(X)) 1131.70/291.60 1131.70/291.60 S is empty. 1131.70/291.60 Rewrite Strategy: FULL 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1131.70/291.60 Transformed a relative TRS into a decreasing-loop problem. 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (2) 1131.70/291.60 Obligation: 1131.70/291.60 Analyzing the following TRS for decreasing loops: 1131.70/291.60 1131.70/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.70/291.60 1131.70/291.60 1131.70/291.60 The TRS R consists of the following rules: 1131.70/291.60 1131.70/291.60 active(nats) -> mark(adx(zeros)) 1131.70/291.60 active(zeros) -> mark(cons(0, zeros)) 1131.70/291.60 active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y))) 1131.70/291.60 active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y)))) 1131.70/291.60 active(hd(cons(X, Y))) -> mark(X) 1131.70/291.60 active(tl(cons(X, Y))) -> mark(Y) 1131.70/291.60 active(adx(X)) -> adx(active(X)) 1131.70/291.60 active(incr(X)) -> incr(active(X)) 1131.70/291.60 active(hd(X)) -> hd(active(X)) 1131.70/291.60 active(tl(X)) -> tl(active(X)) 1131.70/291.60 adx(mark(X)) -> mark(adx(X)) 1131.70/291.60 incr(mark(X)) -> mark(incr(X)) 1131.70/291.60 hd(mark(X)) -> mark(hd(X)) 1131.70/291.60 tl(mark(X)) -> mark(tl(X)) 1131.70/291.60 proper(nats) -> ok(nats) 1131.70/291.60 proper(adx(X)) -> adx(proper(X)) 1131.70/291.60 proper(zeros) -> ok(zeros) 1131.70/291.60 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1131.70/291.60 proper(0) -> ok(0) 1131.70/291.60 proper(incr(X)) -> incr(proper(X)) 1131.70/291.60 proper(s(X)) -> s(proper(X)) 1131.70/291.60 proper(hd(X)) -> hd(proper(X)) 1131.70/291.60 proper(tl(X)) -> tl(proper(X)) 1131.70/291.60 adx(ok(X)) -> ok(adx(X)) 1131.70/291.60 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1131.70/291.60 incr(ok(X)) -> ok(incr(X)) 1131.70/291.60 s(ok(X)) -> ok(s(X)) 1131.70/291.60 hd(ok(X)) -> ok(hd(X)) 1131.70/291.60 tl(ok(X)) -> ok(tl(X)) 1131.70/291.60 top(mark(X)) -> top(proper(X)) 1131.70/291.60 top(ok(X)) -> top(active(X)) 1131.70/291.60 1131.70/291.60 S is empty. 1131.70/291.60 Rewrite Strategy: FULL 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1131.70/291.60 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1131.70/291.60 1131.70/291.60 The rewrite sequence 1131.70/291.60 1131.70/291.60 hd(mark(X)) ->^+ mark(hd(X)) 1131.70/291.60 1131.70/291.60 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1131.70/291.60 1131.70/291.60 The pumping substitution is [X / mark(X)]. 1131.70/291.60 1131.70/291.60 The result substitution is [ ]. 1131.70/291.60 1131.70/291.60 1131.70/291.60 1131.70/291.60 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (4) 1131.70/291.60 Complex Obligation (BEST) 1131.70/291.60 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (5) 1131.70/291.60 Obligation: 1131.70/291.60 Proved the lower bound n^1 for the following obligation: 1131.70/291.60 1131.70/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.70/291.60 1131.70/291.60 1131.70/291.60 The TRS R consists of the following rules: 1131.70/291.60 1131.70/291.60 active(nats) -> mark(adx(zeros)) 1131.70/291.60 active(zeros) -> mark(cons(0, zeros)) 1131.70/291.60 active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y))) 1131.70/291.60 active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y)))) 1131.70/291.60 active(hd(cons(X, Y))) -> mark(X) 1131.70/291.60 active(tl(cons(X, Y))) -> mark(Y) 1131.70/291.60 active(adx(X)) -> adx(active(X)) 1131.70/291.60 active(incr(X)) -> incr(active(X)) 1131.70/291.60 active(hd(X)) -> hd(active(X)) 1131.70/291.60 active(tl(X)) -> tl(active(X)) 1131.70/291.60 adx(mark(X)) -> mark(adx(X)) 1131.70/291.60 incr(mark(X)) -> mark(incr(X)) 1131.70/291.60 hd(mark(X)) -> mark(hd(X)) 1131.70/291.60 tl(mark(X)) -> mark(tl(X)) 1131.70/291.60 proper(nats) -> ok(nats) 1131.70/291.60 proper(adx(X)) -> adx(proper(X)) 1131.70/291.60 proper(zeros) -> ok(zeros) 1131.70/291.60 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1131.70/291.60 proper(0) -> ok(0) 1131.70/291.60 proper(incr(X)) -> incr(proper(X)) 1131.70/291.60 proper(s(X)) -> s(proper(X)) 1131.70/291.60 proper(hd(X)) -> hd(proper(X)) 1131.70/291.60 proper(tl(X)) -> tl(proper(X)) 1131.70/291.60 adx(ok(X)) -> ok(adx(X)) 1131.70/291.60 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1131.70/291.60 incr(ok(X)) -> ok(incr(X)) 1131.70/291.60 s(ok(X)) -> ok(s(X)) 1131.70/291.60 hd(ok(X)) -> ok(hd(X)) 1131.70/291.60 tl(ok(X)) -> ok(tl(X)) 1131.70/291.60 top(mark(X)) -> top(proper(X)) 1131.70/291.60 top(ok(X)) -> top(active(X)) 1131.70/291.60 1131.70/291.60 S is empty. 1131.70/291.60 Rewrite Strategy: FULL 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (6) LowerBoundPropagationProof (FINISHED) 1131.70/291.60 Propagated lower bound. 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (7) 1131.70/291.60 BOUNDS(n^1, INF) 1131.70/291.60 1131.70/291.60 ---------------------------------------- 1131.70/291.60 1131.70/291.60 (8) 1131.70/291.60 Obligation: 1131.70/291.60 Analyzing the following TRS for decreasing loops: 1131.70/291.60 1131.70/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.70/291.60 1131.70/291.60 1131.70/291.60 The TRS R consists of the following rules: 1131.70/291.60 1131.70/291.60 active(nats) -> mark(adx(zeros)) 1131.70/291.60 active(zeros) -> mark(cons(0, zeros)) 1131.70/291.60 active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y))) 1131.70/291.60 active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y)))) 1131.70/291.60 active(hd(cons(X, Y))) -> mark(X) 1131.70/291.60 active(tl(cons(X, Y))) -> mark(Y) 1131.70/291.60 active(adx(X)) -> adx(active(X)) 1131.70/291.60 active(incr(X)) -> incr(active(X)) 1131.70/291.60 active(hd(X)) -> hd(active(X)) 1131.70/291.60 active(tl(X)) -> tl(active(X)) 1131.70/291.60 adx(mark(X)) -> mark(adx(X)) 1131.70/291.60 incr(mark(X)) -> mark(incr(X)) 1131.70/291.60 hd(mark(X)) -> mark(hd(X)) 1131.70/291.60 tl(mark(X)) -> mark(tl(X)) 1131.70/291.60 proper(nats) -> ok(nats) 1131.70/291.60 proper(adx(X)) -> adx(proper(X)) 1131.70/291.60 proper(zeros) -> ok(zeros) 1131.70/291.60 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1131.70/291.60 proper(0) -> ok(0) 1131.70/291.60 proper(incr(X)) -> incr(proper(X)) 1131.70/291.60 proper(s(X)) -> s(proper(X)) 1131.70/291.60 proper(hd(X)) -> hd(proper(X)) 1131.70/291.60 proper(tl(X)) -> tl(proper(X)) 1131.70/291.60 adx(ok(X)) -> ok(adx(X)) 1131.70/291.60 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1131.70/291.60 incr(ok(X)) -> ok(incr(X)) 1131.70/291.60 s(ok(X)) -> ok(s(X)) 1131.70/291.60 hd(ok(X)) -> ok(hd(X)) 1131.70/291.60 tl(ok(X)) -> ok(tl(X)) 1131.70/291.60 top(mark(X)) -> top(proper(X)) 1131.70/291.60 top(ok(X)) -> top(active(X)) 1131.70/291.60 1131.70/291.60 S is empty. 1131.70/291.60 Rewrite Strategy: FULL 1131.83/291.66 EOF