1138.39/291.93 WORST_CASE(Omega(n^1), ?) 1141.41/292.65 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1141.41/292.65 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1141.41/292.65 1141.41/292.65 1141.41/292.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1141.41/292.65 1141.41/292.65 (0) CpxTRS 1141.41/292.65 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1141.41/292.65 (2) TRS for Loop Detection 1141.41/292.65 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1141.41/292.65 (4) BEST 1141.41/292.65 (5) proven lower bound 1141.41/292.65 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1141.41/292.65 (7) BOUNDS(n^1, INF) 1141.41/292.65 (8) TRS for Loop Detection 1141.41/292.65 1141.41/292.65 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (0) 1141.41/292.65 Obligation: 1141.41/292.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1141.41/292.65 1141.41/292.65 1141.41/292.65 The TRS R consists of the following rules: 1141.41/292.65 1141.41/292.65 active(incr(nil)) -> mark(nil) 1141.41/292.65 active(incr(cons(X, L))) -> mark(cons(s(X), incr(L))) 1141.41/292.65 active(adx(nil)) -> mark(nil) 1141.41/292.65 active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L)))) 1141.41/292.65 active(nats) -> mark(adx(zeros)) 1141.41/292.65 active(zeros) -> mark(cons(0, zeros)) 1141.41/292.65 active(head(cons(X, L))) -> mark(X) 1141.41/292.65 active(tail(cons(X, L))) -> mark(L) 1141.41/292.65 active(incr(X)) -> incr(active(X)) 1141.41/292.65 active(cons(X1, X2)) -> cons(active(X1), X2) 1141.41/292.65 active(s(X)) -> s(active(X)) 1141.41/292.65 active(adx(X)) -> adx(active(X)) 1141.41/292.65 active(head(X)) -> head(active(X)) 1141.41/292.65 active(tail(X)) -> tail(active(X)) 1141.41/292.65 incr(mark(X)) -> mark(incr(X)) 1141.41/292.65 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1141.41/292.65 s(mark(X)) -> mark(s(X)) 1141.41/292.65 adx(mark(X)) -> mark(adx(X)) 1141.41/292.65 head(mark(X)) -> mark(head(X)) 1141.41/292.65 tail(mark(X)) -> mark(tail(X)) 1141.41/292.65 proper(incr(X)) -> incr(proper(X)) 1141.41/292.65 proper(nil) -> ok(nil) 1141.41/292.65 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1141.41/292.65 proper(s(X)) -> s(proper(X)) 1141.41/292.65 proper(adx(X)) -> adx(proper(X)) 1141.41/292.65 proper(nats) -> ok(nats) 1141.41/292.65 proper(zeros) -> ok(zeros) 1141.41/292.65 proper(0) -> ok(0) 1141.41/292.65 proper(head(X)) -> head(proper(X)) 1141.41/292.65 proper(tail(X)) -> tail(proper(X)) 1141.41/292.65 incr(ok(X)) -> ok(incr(X)) 1141.41/292.65 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1141.41/292.65 s(ok(X)) -> ok(s(X)) 1141.41/292.65 adx(ok(X)) -> ok(adx(X)) 1141.41/292.65 head(ok(X)) -> ok(head(X)) 1141.41/292.65 tail(ok(X)) -> ok(tail(X)) 1141.41/292.65 top(mark(X)) -> top(proper(X)) 1141.41/292.65 top(ok(X)) -> top(active(X)) 1141.41/292.65 1141.41/292.65 S is empty. 1141.41/292.65 Rewrite Strategy: FULL 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1141.41/292.65 Transformed a relative TRS into a decreasing-loop problem. 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (2) 1141.41/292.65 Obligation: 1141.41/292.65 Analyzing the following TRS for decreasing loops: 1141.41/292.65 1141.41/292.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1141.41/292.65 1141.41/292.65 1141.41/292.65 The TRS R consists of the following rules: 1141.41/292.65 1141.41/292.65 active(incr(nil)) -> mark(nil) 1141.41/292.65 active(incr(cons(X, L))) -> mark(cons(s(X), incr(L))) 1141.41/292.65 active(adx(nil)) -> mark(nil) 1141.41/292.65 active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L)))) 1141.41/292.65 active(nats) -> mark(adx(zeros)) 1141.41/292.65 active(zeros) -> mark(cons(0, zeros)) 1141.41/292.65 active(head(cons(X, L))) -> mark(X) 1141.41/292.65 active(tail(cons(X, L))) -> mark(L) 1141.41/292.65 active(incr(X)) -> incr(active(X)) 1141.41/292.65 active(cons(X1, X2)) -> cons(active(X1), X2) 1141.41/292.65 active(s(X)) -> s(active(X)) 1141.41/292.65 active(adx(X)) -> adx(active(X)) 1141.41/292.65 active(head(X)) -> head(active(X)) 1141.41/292.65 active(tail(X)) -> tail(active(X)) 1141.41/292.65 incr(mark(X)) -> mark(incr(X)) 1141.41/292.65 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1141.41/292.65 s(mark(X)) -> mark(s(X)) 1141.41/292.65 adx(mark(X)) -> mark(adx(X)) 1141.41/292.65 head(mark(X)) -> mark(head(X)) 1141.41/292.65 tail(mark(X)) -> mark(tail(X)) 1141.41/292.65 proper(incr(X)) -> incr(proper(X)) 1141.41/292.65 proper(nil) -> ok(nil) 1141.41/292.65 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1141.41/292.65 proper(s(X)) -> s(proper(X)) 1141.41/292.65 proper(adx(X)) -> adx(proper(X)) 1141.41/292.65 proper(nats) -> ok(nats) 1141.41/292.65 proper(zeros) -> ok(zeros) 1141.41/292.65 proper(0) -> ok(0) 1141.41/292.65 proper(head(X)) -> head(proper(X)) 1141.41/292.65 proper(tail(X)) -> tail(proper(X)) 1141.41/292.65 incr(ok(X)) -> ok(incr(X)) 1141.41/292.65 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1141.41/292.65 s(ok(X)) -> ok(s(X)) 1141.41/292.65 adx(ok(X)) -> ok(adx(X)) 1141.41/292.65 head(ok(X)) -> ok(head(X)) 1141.41/292.65 tail(ok(X)) -> ok(tail(X)) 1141.41/292.65 top(mark(X)) -> top(proper(X)) 1141.41/292.65 top(ok(X)) -> top(active(X)) 1141.41/292.65 1141.41/292.65 S is empty. 1141.41/292.65 Rewrite Strategy: FULL 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1141.41/292.65 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1141.41/292.65 1141.41/292.65 The rewrite sequence 1141.41/292.65 1141.41/292.65 head(ok(X)) ->^+ ok(head(X)) 1141.41/292.65 1141.41/292.65 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1141.41/292.65 1141.41/292.65 The pumping substitution is [X / ok(X)]. 1141.41/292.65 1141.41/292.65 The result substitution is [ ]. 1141.41/292.65 1141.41/292.65 1141.41/292.65 1141.41/292.65 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (4) 1141.41/292.65 Complex Obligation (BEST) 1141.41/292.65 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (5) 1141.41/292.65 Obligation: 1141.41/292.65 Proved the lower bound n^1 for the following obligation: 1141.41/292.65 1141.41/292.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1141.41/292.65 1141.41/292.65 1141.41/292.65 The TRS R consists of the following rules: 1141.41/292.65 1141.41/292.65 active(incr(nil)) -> mark(nil) 1141.41/292.65 active(incr(cons(X, L))) -> mark(cons(s(X), incr(L))) 1141.41/292.65 active(adx(nil)) -> mark(nil) 1141.41/292.65 active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L)))) 1141.41/292.65 active(nats) -> mark(adx(zeros)) 1141.41/292.65 active(zeros) -> mark(cons(0, zeros)) 1141.41/292.65 active(head(cons(X, L))) -> mark(X) 1141.41/292.65 active(tail(cons(X, L))) -> mark(L) 1141.41/292.65 active(incr(X)) -> incr(active(X)) 1141.41/292.65 active(cons(X1, X2)) -> cons(active(X1), X2) 1141.41/292.65 active(s(X)) -> s(active(X)) 1141.41/292.65 active(adx(X)) -> adx(active(X)) 1141.41/292.65 active(head(X)) -> head(active(X)) 1141.41/292.65 active(tail(X)) -> tail(active(X)) 1141.41/292.65 incr(mark(X)) -> mark(incr(X)) 1141.41/292.65 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1141.41/292.65 s(mark(X)) -> mark(s(X)) 1141.41/292.65 adx(mark(X)) -> mark(adx(X)) 1141.41/292.65 head(mark(X)) -> mark(head(X)) 1141.41/292.65 tail(mark(X)) -> mark(tail(X)) 1141.41/292.65 proper(incr(X)) -> incr(proper(X)) 1141.41/292.65 proper(nil) -> ok(nil) 1141.41/292.65 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1141.41/292.65 proper(s(X)) -> s(proper(X)) 1141.41/292.65 proper(adx(X)) -> adx(proper(X)) 1141.41/292.65 proper(nats) -> ok(nats) 1141.41/292.65 proper(zeros) -> ok(zeros) 1141.41/292.65 proper(0) -> ok(0) 1141.41/292.65 proper(head(X)) -> head(proper(X)) 1141.41/292.65 proper(tail(X)) -> tail(proper(X)) 1141.41/292.65 incr(ok(X)) -> ok(incr(X)) 1141.41/292.65 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1141.41/292.65 s(ok(X)) -> ok(s(X)) 1141.41/292.65 adx(ok(X)) -> ok(adx(X)) 1141.41/292.65 head(ok(X)) -> ok(head(X)) 1141.41/292.65 tail(ok(X)) -> ok(tail(X)) 1141.41/292.65 top(mark(X)) -> top(proper(X)) 1141.41/292.65 top(ok(X)) -> top(active(X)) 1141.41/292.65 1141.41/292.65 S is empty. 1141.41/292.65 Rewrite Strategy: FULL 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (6) LowerBoundPropagationProof (FINISHED) 1141.41/292.65 Propagated lower bound. 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (7) 1141.41/292.65 BOUNDS(n^1, INF) 1141.41/292.65 1141.41/292.65 ---------------------------------------- 1141.41/292.65 1141.41/292.65 (8) 1141.41/292.65 Obligation: 1141.41/292.65 Analyzing the following TRS for decreasing loops: 1141.41/292.65 1141.41/292.65 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1141.41/292.65 1141.41/292.65 1141.41/292.65 The TRS R consists of the following rules: 1141.41/292.65 1141.41/292.65 active(incr(nil)) -> mark(nil) 1141.41/292.65 active(incr(cons(X, L))) -> mark(cons(s(X), incr(L))) 1141.41/292.65 active(adx(nil)) -> mark(nil) 1141.41/292.65 active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L)))) 1141.41/292.65 active(nats) -> mark(adx(zeros)) 1141.41/292.65 active(zeros) -> mark(cons(0, zeros)) 1141.41/292.65 active(head(cons(X, L))) -> mark(X) 1141.41/292.65 active(tail(cons(X, L))) -> mark(L) 1141.41/292.65 active(incr(X)) -> incr(active(X)) 1141.41/292.65 active(cons(X1, X2)) -> cons(active(X1), X2) 1141.41/292.65 active(s(X)) -> s(active(X)) 1141.41/292.65 active(adx(X)) -> adx(active(X)) 1141.41/292.65 active(head(X)) -> head(active(X)) 1141.41/292.65 active(tail(X)) -> tail(active(X)) 1141.41/292.65 incr(mark(X)) -> mark(incr(X)) 1141.41/292.65 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1141.41/292.65 s(mark(X)) -> mark(s(X)) 1141.41/292.65 adx(mark(X)) -> mark(adx(X)) 1141.41/292.65 head(mark(X)) -> mark(head(X)) 1141.41/292.65 tail(mark(X)) -> mark(tail(X)) 1141.41/292.65 proper(incr(X)) -> incr(proper(X)) 1141.41/292.65 proper(nil) -> ok(nil) 1141.41/292.65 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1141.41/292.65 proper(s(X)) -> s(proper(X)) 1141.41/292.65 proper(adx(X)) -> adx(proper(X)) 1141.41/292.65 proper(nats) -> ok(nats) 1141.41/292.65 proper(zeros) -> ok(zeros) 1141.41/292.65 proper(0) -> ok(0) 1141.41/292.65 proper(head(X)) -> head(proper(X)) 1141.41/292.65 proper(tail(X)) -> tail(proper(X)) 1141.41/292.65 incr(ok(X)) -> ok(incr(X)) 1141.41/292.65 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1141.41/292.65 s(ok(X)) -> ok(s(X)) 1141.41/292.65 adx(ok(X)) -> ok(adx(X)) 1141.41/292.65 head(ok(X)) -> ok(head(X)) 1141.41/292.65 tail(ok(X)) -> ok(tail(X)) 1141.41/292.65 top(mark(X)) -> top(proper(X)) 1141.41/292.65 top(ok(X)) -> top(active(X)) 1141.41/292.65 1141.41/292.65 S is empty. 1141.41/292.65 Rewrite Strategy: FULL 1141.52/292.74 EOF