375.43/291.51 WORST_CASE(Omega(n^1), ?) 375.43/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 375.43/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 375.43/291.52 375.43/291.52 375.43/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 375.43/291.52 375.43/291.52 (0) CpxTRS 375.43/291.52 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 375.43/291.52 (2) TRS for Loop Detection 375.43/291.52 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 375.43/291.52 (4) BEST 375.43/291.52 (5) proven lower bound 375.43/291.52 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 375.43/291.52 (7) BOUNDS(n^1, INF) 375.43/291.52 (8) TRS for Loop Detection 375.43/291.52 375.43/291.52 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (0) 375.43/291.52 Obligation: 375.43/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 375.43/291.52 375.43/291.52 375.43/291.52 The TRS R consists of the following rules: 375.43/291.52 375.43/291.52 active(primes) -> mark(sieve(from(s(s(0))))) 375.43/291.52 active(from(X)) -> mark(cons(X, from(s(X)))) 375.43/291.52 active(head(cons(X, Y))) -> mark(X) 375.43/291.52 active(tail(cons(X, Y))) -> mark(Y) 375.43/291.52 active(if(true, X, Y)) -> mark(X) 375.43/291.52 active(if(false, X, Y)) -> mark(Y) 375.43/291.52 active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))) 375.43/291.52 active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y)))) 375.43/291.52 active(sieve(X)) -> sieve(active(X)) 375.43/291.52 active(from(X)) -> from(active(X)) 375.43/291.52 active(s(X)) -> s(active(X)) 375.43/291.52 active(cons(X1, X2)) -> cons(active(X1), X2) 375.43/291.52 active(head(X)) -> head(active(X)) 375.43/291.52 active(tail(X)) -> tail(active(X)) 375.43/291.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 375.43/291.52 active(filter(X1, X2)) -> filter(active(X1), X2) 375.43/291.52 active(filter(X1, X2)) -> filter(X1, active(X2)) 375.43/291.52 active(divides(X1, X2)) -> divides(active(X1), X2) 375.43/291.52 active(divides(X1, X2)) -> divides(X1, active(X2)) 375.43/291.52 sieve(mark(X)) -> mark(sieve(X)) 375.43/291.52 from(mark(X)) -> mark(from(X)) 375.43/291.52 s(mark(X)) -> mark(s(X)) 375.43/291.52 cons(mark(X1), X2) -> mark(cons(X1, X2)) 375.43/291.52 head(mark(X)) -> mark(head(X)) 375.43/291.52 tail(mark(X)) -> mark(tail(X)) 375.43/291.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 375.43/291.52 filter(mark(X1), X2) -> mark(filter(X1, X2)) 375.43/291.52 filter(X1, mark(X2)) -> mark(filter(X1, X2)) 375.43/291.52 divides(mark(X1), X2) -> mark(divides(X1, X2)) 375.43/291.52 divides(X1, mark(X2)) -> mark(divides(X1, X2)) 375.43/291.52 proper(primes) -> ok(primes) 375.43/291.52 proper(sieve(X)) -> sieve(proper(X)) 375.43/291.52 proper(from(X)) -> from(proper(X)) 375.43/291.52 proper(s(X)) -> s(proper(X)) 375.43/291.52 proper(0) -> ok(0) 375.43/291.52 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 375.43/291.52 proper(head(X)) -> head(proper(X)) 375.43/291.52 proper(tail(X)) -> tail(proper(X)) 375.43/291.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 375.43/291.52 proper(true) -> ok(true) 375.43/291.52 proper(false) -> ok(false) 375.43/291.52 proper(filter(X1, X2)) -> filter(proper(X1), proper(X2)) 375.43/291.52 proper(divides(X1, X2)) -> divides(proper(X1), proper(X2)) 375.43/291.52 sieve(ok(X)) -> ok(sieve(X)) 375.43/291.52 from(ok(X)) -> ok(from(X)) 375.43/291.52 s(ok(X)) -> ok(s(X)) 375.43/291.52 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 375.43/291.52 head(ok(X)) -> ok(head(X)) 375.43/291.52 tail(ok(X)) -> ok(tail(X)) 375.43/291.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 375.43/291.52 filter(ok(X1), ok(X2)) -> ok(filter(X1, X2)) 375.43/291.52 divides(ok(X1), ok(X2)) -> ok(divides(X1, X2)) 375.43/291.52 top(mark(X)) -> top(proper(X)) 375.43/291.52 top(ok(X)) -> top(active(X)) 375.43/291.52 375.43/291.52 S is empty. 375.43/291.52 Rewrite Strategy: FULL 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 375.43/291.52 Transformed a relative TRS into a decreasing-loop problem. 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (2) 375.43/291.52 Obligation: 375.43/291.52 Analyzing the following TRS for decreasing loops: 375.43/291.52 375.43/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 375.43/291.52 375.43/291.52 375.43/291.52 The TRS R consists of the following rules: 375.43/291.52 375.43/291.52 active(primes) -> mark(sieve(from(s(s(0))))) 375.43/291.52 active(from(X)) -> mark(cons(X, from(s(X)))) 375.43/291.52 active(head(cons(X, Y))) -> mark(X) 375.43/291.52 active(tail(cons(X, Y))) -> mark(Y) 375.43/291.52 active(if(true, X, Y)) -> mark(X) 375.43/291.52 active(if(false, X, Y)) -> mark(Y) 375.43/291.52 active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))) 375.43/291.52 active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y)))) 375.43/291.52 active(sieve(X)) -> sieve(active(X)) 375.43/291.52 active(from(X)) -> from(active(X)) 375.43/291.52 active(s(X)) -> s(active(X)) 375.43/291.52 active(cons(X1, X2)) -> cons(active(X1), X2) 375.43/291.52 active(head(X)) -> head(active(X)) 375.43/291.52 active(tail(X)) -> tail(active(X)) 375.43/291.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 375.43/291.52 active(filter(X1, X2)) -> filter(active(X1), X2) 375.43/291.52 active(filter(X1, X2)) -> filter(X1, active(X2)) 375.43/291.52 active(divides(X1, X2)) -> divides(active(X1), X2) 375.43/291.52 active(divides(X1, X2)) -> divides(X1, active(X2)) 375.43/291.52 sieve(mark(X)) -> mark(sieve(X)) 375.43/291.52 from(mark(X)) -> mark(from(X)) 375.43/291.52 s(mark(X)) -> mark(s(X)) 375.43/291.52 cons(mark(X1), X2) -> mark(cons(X1, X2)) 375.43/291.52 head(mark(X)) -> mark(head(X)) 375.43/291.52 tail(mark(X)) -> mark(tail(X)) 375.43/291.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 375.43/291.52 filter(mark(X1), X2) -> mark(filter(X1, X2)) 375.43/291.52 filter(X1, mark(X2)) -> mark(filter(X1, X2)) 375.43/291.52 divides(mark(X1), X2) -> mark(divides(X1, X2)) 375.43/291.52 divides(X1, mark(X2)) -> mark(divides(X1, X2)) 375.43/291.52 proper(primes) -> ok(primes) 375.43/291.52 proper(sieve(X)) -> sieve(proper(X)) 375.43/291.52 proper(from(X)) -> from(proper(X)) 375.43/291.52 proper(s(X)) -> s(proper(X)) 375.43/291.52 proper(0) -> ok(0) 375.43/291.52 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 375.43/291.52 proper(head(X)) -> head(proper(X)) 375.43/291.52 proper(tail(X)) -> tail(proper(X)) 375.43/291.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 375.43/291.52 proper(true) -> ok(true) 375.43/291.52 proper(false) -> ok(false) 375.43/291.52 proper(filter(X1, X2)) -> filter(proper(X1), proper(X2)) 375.43/291.52 proper(divides(X1, X2)) -> divides(proper(X1), proper(X2)) 375.43/291.52 sieve(ok(X)) -> ok(sieve(X)) 375.43/291.52 from(ok(X)) -> ok(from(X)) 375.43/291.52 s(ok(X)) -> ok(s(X)) 375.43/291.52 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 375.43/291.52 head(ok(X)) -> ok(head(X)) 375.43/291.52 tail(ok(X)) -> ok(tail(X)) 375.43/291.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 375.43/291.52 filter(ok(X1), ok(X2)) -> ok(filter(X1, X2)) 375.43/291.52 divides(ok(X1), ok(X2)) -> ok(divides(X1, X2)) 375.43/291.52 top(mark(X)) -> top(proper(X)) 375.43/291.52 top(ok(X)) -> top(active(X)) 375.43/291.52 375.43/291.52 S is empty. 375.43/291.52 Rewrite Strategy: FULL 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (3) DecreasingLoopProof (LOWER BOUND(ID)) 375.43/291.52 The following loop(s) give(s) rise to the lower bound Omega(n^1): 375.43/291.52 375.43/291.52 The rewrite sequence 375.43/291.52 375.43/291.52 tail(ok(X)) ->^+ ok(tail(X)) 375.43/291.52 375.43/291.52 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 375.43/291.52 375.43/291.52 The pumping substitution is [X / ok(X)]. 375.43/291.52 375.43/291.52 The result substitution is [ ]. 375.43/291.52 375.43/291.52 375.43/291.52 375.43/291.52 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (4) 375.43/291.52 Complex Obligation (BEST) 375.43/291.52 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (5) 375.43/291.52 Obligation: 375.43/291.52 Proved the lower bound n^1 for the following obligation: 375.43/291.52 375.43/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 375.43/291.52 375.43/291.52 375.43/291.52 The TRS R consists of the following rules: 375.43/291.52 375.43/291.52 active(primes) -> mark(sieve(from(s(s(0))))) 375.43/291.52 active(from(X)) -> mark(cons(X, from(s(X)))) 375.43/291.52 active(head(cons(X, Y))) -> mark(X) 375.43/291.52 active(tail(cons(X, Y))) -> mark(Y) 375.43/291.52 active(if(true, X, Y)) -> mark(X) 375.43/291.52 active(if(false, X, Y)) -> mark(Y) 375.43/291.52 active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))) 375.43/291.52 active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y)))) 375.43/291.52 active(sieve(X)) -> sieve(active(X)) 375.43/291.52 active(from(X)) -> from(active(X)) 375.43/291.52 active(s(X)) -> s(active(X)) 375.43/291.52 active(cons(X1, X2)) -> cons(active(X1), X2) 375.43/291.52 active(head(X)) -> head(active(X)) 375.43/291.52 active(tail(X)) -> tail(active(X)) 375.43/291.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 375.43/291.52 active(filter(X1, X2)) -> filter(active(X1), X2) 375.43/291.52 active(filter(X1, X2)) -> filter(X1, active(X2)) 375.43/291.52 active(divides(X1, X2)) -> divides(active(X1), X2) 375.43/291.52 active(divides(X1, X2)) -> divides(X1, active(X2)) 375.43/291.52 sieve(mark(X)) -> mark(sieve(X)) 375.43/291.52 from(mark(X)) -> mark(from(X)) 375.43/291.52 s(mark(X)) -> mark(s(X)) 375.43/291.52 cons(mark(X1), X2) -> mark(cons(X1, X2)) 375.43/291.52 head(mark(X)) -> mark(head(X)) 375.43/291.52 tail(mark(X)) -> mark(tail(X)) 375.43/291.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 375.43/291.52 filter(mark(X1), X2) -> mark(filter(X1, X2)) 375.43/291.52 filter(X1, mark(X2)) -> mark(filter(X1, X2)) 375.43/291.52 divides(mark(X1), X2) -> mark(divides(X1, X2)) 375.43/291.52 divides(X1, mark(X2)) -> mark(divides(X1, X2)) 375.43/291.52 proper(primes) -> ok(primes) 375.43/291.52 proper(sieve(X)) -> sieve(proper(X)) 375.43/291.52 proper(from(X)) -> from(proper(X)) 375.43/291.52 proper(s(X)) -> s(proper(X)) 375.43/291.52 proper(0) -> ok(0) 375.43/291.52 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 375.43/291.52 proper(head(X)) -> head(proper(X)) 375.43/291.52 proper(tail(X)) -> tail(proper(X)) 375.43/291.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 375.43/291.52 proper(true) -> ok(true) 375.43/291.52 proper(false) -> ok(false) 375.43/291.52 proper(filter(X1, X2)) -> filter(proper(X1), proper(X2)) 375.43/291.52 proper(divides(X1, X2)) -> divides(proper(X1), proper(X2)) 375.43/291.52 sieve(ok(X)) -> ok(sieve(X)) 375.43/291.52 from(ok(X)) -> ok(from(X)) 375.43/291.52 s(ok(X)) -> ok(s(X)) 375.43/291.52 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 375.43/291.52 head(ok(X)) -> ok(head(X)) 375.43/291.52 tail(ok(X)) -> ok(tail(X)) 375.43/291.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 375.43/291.52 filter(ok(X1), ok(X2)) -> ok(filter(X1, X2)) 375.43/291.52 divides(ok(X1), ok(X2)) -> ok(divides(X1, X2)) 375.43/291.52 top(mark(X)) -> top(proper(X)) 375.43/291.52 top(ok(X)) -> top(active(X)) 375.43/291.52 375.43/291.52 S is empty. 375.43/291.52 Rewrite Strategy: FULL 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (6) LowerBoundPropagationProof (FINISHED) 375.43/291.52 Propagated lower bound. 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (7) 375.43/291.52 BOUNDS(n^1, INF) 375.43/291.52 375.43/291.52 ---------------------------------------- 375.43/291.52 375.43/291.52 (8) 375.43/291.52 Obligation: 375.43/291.52 Analyzing the following TRS for decreasing loops: 375.43/291.52 375.43/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 375.43/291.52 375.43/291.52 375.43/291.52 The TRS R consists of the following rules: 375.43/291.52 375.43/291.52 active(primes) -> mark(sieve(from(s(s(0))))) 375.43/291.52 active(from(X)) -> mark(cons(X, from(s(X)))) 375.43/291.52 active(head(cons(X, Y))) -> mark(X) 375.43/291.52 active(tail(cons(X, Y))) -> mark(Y) 375.43/291.52 active(if(true, X, Y)) -> mark(X) 375.43/291.52 active(if(false, X, Y)) -> mark(Y) 375.43/291.52 active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))) 375.43/291.52 active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y)))) 375.43/291.52 active(sieve(X)) -> sieve(active(X)) 375.43/291.52 active(from(X)) -> from(active(X)) 375.43/291.52 active(s(X)) -> s(active(X)) 375.43/291.52 active(cons(X1, X2)) -> cons(active(X1), X2) 375.43/291.52 active(head(X)) -> head(active(X)) 375.43/291.52 active(tail(X)) -> tail(active(X)) 375.43/291.52 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 375.43/291.52 active(filter(X1, X2)) -> filter(active(X1), X2) 375.43/291.52 active(filter(X1, X2)) -> filter(X1, active(X2)) 375.43/291.52 active(divides(X1, X2)) -> divides(active(X1), X2) 375.43/291.52 active(divides(X1, X2)) -> divides(X1, active(X2)) 375.43/291.52 sieve(mark(X)) -> mark(sieve(X)) 375.43/291.52 from(mark(X)) -> mark(from(X)) 375.43/291.52 s(mark(X)) -> mark(s(X)) 375.43/291.52 cons(mark(X1), X2) -> mark(cons(X1, X2)) 375.43/291.52 head(mark(X)) -> mark(head(X)) 375.43/291.52 tail(mark(X)) -> mark(tail(X)) 375.43/291.52 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 375.43/291.52 filter(mark(X1), X2) -> mark(filter(X1, X2)) 375.43/291.52 filter(X1, mark(X2)) -> mark(filter(X1, X2)) 375.43/291.52 divides(mark(X1), X2) -> mark(divides(X1, X2)) 375.43/291.52 divides(X1, mark(X2)) -> mark(divides(X1, X2)) 375.43/291.52 proper(primes) -> ok(primes) 375.43/291.52 proper(sieve(X)) -> sieve(proper(X)) 375.43/291.52 proper(from(X)) -> from(proper(X)) 375.43/291.52 proper(s(X)) -> s(proper(X)) 375.43/291.52 proper(0) -> ok(0) 375.43/291.52 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 375.43/291.52 proper(head(X)) -> head(proper(X)) 375.43/291.52 proper(tail(X)) -> tail(proper(X)) 375.43/291.52 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 375.43/291.52 proper(true) -> ok(true) 375.43/291.52 proper(false) -> ok(false) 375.43/291.52 proper(filter(X1, X2)) -> filter(proper(X1), proper(X2)) 375.43/291.52 proper(divides(X1, X2)) -> divides(proper(X1), proper(X2)) 375.43/291.52 sieve(ok(X)) -> ok(sieve(X)) 375.43/291.52 from(ok(X)) -> ok(from(X)) 375.43/291.52 s(ok(X)) -> ok(s(X)) 375.43/291.52 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 375.43/291.52 head(ok(X)) -> ok(head(X)) 375.43/291.52 tail(ok(X)) -> ok(tail(X)) 375.43/291.52 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 375.43/291.52 filter(ok(X1), ok(X2)) -> ok(filter(X1, X2)) 375.43/291.52 divides(ok(X1), ok(X2)) -> ok(divides(X1, X2)) 375.43/291.52 top(mark(X)) -> top(proper(X)) 375.43/291.52 top(ok(X)) -> top(active(X)) 375.43/291.52 375.43/291.52 S is empty. 375.43/291.52 Rewrite Strategy: FULL 375.51/291.57 EOF