/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 372 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_inc(x_1)) -> inc(encArg(x_1))
   encArg(cons_sumList(x_1, x_2)) -> sumList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_isEmpty(x_1) -> isEmpty(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_nil -> nil
   encode_true -> true
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_inc(x_1) -> inc(encArg(x_1))
   encode_sumList(x_1, x_2) -> sumList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encode_sum(x_1) -> sum(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0)

The (relative) TRS S consists of the following rules:

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_inc(x_1)) -> inc(encArg(x_1))
   encArg(cons_sumList(x_1, x_2)) -> sumList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_isEmpty(x_1) -> isEmpty(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_nil -> nil
   encode_true -> true
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_inc(x_1) -> inc(encArg(x_1))
   encode_sumList(x_1, x_2) -> sumList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0)

The (relative) TRS S consists of the following rules:

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_inc(x_1)) -> inc(encArg(x_1))
   encArg(cons_sumList(x_1, x_2)) -> sumList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_isEmpty(x_1) -> isEmpty(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_nil -> nil
   encode_true -> true
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_inc(x_1) -> inc(encArg(x_1))
   encode_sumList(x_1, x_2) -> sumList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0)

The (relative) TRS S consists of the following rules:

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_inc(x_1)) -> inc(encArg(x_1))
   encArg(cons_sumList(x_1, x_2)) -> sumList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_isEmpty(x_1) -> isEmpty(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_nil -> nil
   encode_true -> true
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_inc(x_1) -> inc(encArg(x_1))
   encode_sumList(x_1, x_2) -> sumList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

The rewrite sequence

p(s(s(x))) ->^+ s(p(s(x)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / s(x)].

The result substitution is [ ].




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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0)

The (relative) TRS S consists of the following rules:

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_inc(x_1)) -> inc(encArg(x_1))
   encArg(cons_sumList(x_1, x_2)) -> sumList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_isEmpty(x_1) -> isEmpty(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_nil -> nil
   encode_true -> true
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_inc(x_1) -> inc(encArg(x_1))
   encode_sumList(x_1, x_2) -> sumList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0)

The (relative) TRS S consists of the following rules:

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_inc(x_1)) -> inc(encArg(x_1))
   encArg(cons_sumList(x_1, x_2)) -> sumList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_isEmpty(x_1) -> isEmpty(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_nil -> nil
   encode_true -> true
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_inc(x_1) -> inc(encArg(x_1))
   encode_sumList(x_1, x_2) -> sumList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST