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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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(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:

   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   bin2s(nil) -> 0
   bin2s(cons(x, xs)) -> bin2ss(x, xs)
   bin2ss(x, nil) -> x
   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   log(0) -> 0
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(half(s(s(x)))))
   more(nil) -> nil
   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
   if1(false, x, y, lists) -> s2bin2(x, lists)
   s2bin2(x, nil) -> bug_list_not
   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
   if2(true, x, xs, ys) -> xs
   if2(false, x, xs, ys) -> s2bin2(x, ys)

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(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:

   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   bin2s(nil) -> 0
   bin2s(cons(x, xs)) -> bin2ss(x, xs)
   bin2ss(x, nil) -> x
   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   log(0) -> 0
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(half(s(s(x)))))
   more(nil) -> nil
   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
   if1(false, x, y, lists) -> s2bin2(x, lists)
   s2bin2(x, nil) -> bug_list_not
   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
   if2(true, x, xs, ys) -> xs
   if2(false, x, xs, ys) -> s2bin2(x, ys)

S is empty.
Rewrite Strategy: FULL
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(3) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

bin2ss(x, cons(0, xs)) ->^+ bin2ss(double(x), 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 [x / double(x)].




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(4)
Complex Obligation (BEST)

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(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:

   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   bin2s(nil) -> 0
   bin2s(cons(x, xs)) -> bin2ss(x, xs)
   bin2ss(x, nil) -> x
   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   log(0) -> 0
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(half(s(s(x)))))
   more(nil) -> nil
   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
   if1(false, x, y, lists) -> s2bin2(x, lists)
   s2bin2(x, nil) -> bug_list_not
   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
   if2(true, x, xs, ys) -> xs
   if2(false, x, xs, ys) -> s2bin2(x, ys)

S is empty.
Rewrite Strategy: FULL
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(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:

   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   bin2s(nil) -> 0
   bin2s(cons(x, xs)) -> bin2ss(x, xs)
   bin2ss(x, nil) -> x
   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   log(0) -> 0
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(half(s(s(x)))))
   more(nil) -> nil
   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
   if1(false, x, y, lists) -> s2bin2(x, lists)
   s2bin2(x, nil) -> bug_list_not
   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
   if2(true, x, xs, ys) -> xs
   if2(false, x, xs, ys) -> s2bin2(x, ys)

S is empty.
Rewrite Strategy: FULL