/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0
   sum(cons(0, xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(2)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0
   sum(cons(0, xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

sum(cons(0, xs)) ->^+ sum(xs)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [xs / cons(0, xs)].

The result substitution is [ ].




----------------------------------------

(4)
Complex Obligation (BEST)

----------------------------------------

(5)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0
   sum(cons(0, xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(7)
BOUNDS(n^1, INF)

----------------------------------------

(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0
   sum(cons(0, xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)

S is empty.
Rewrite Strategy: FULL