/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 270 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:

   min(0, y) -> 0
   min(s(x), 0) -> 0
   min(s(x), s(y)) -> min(x, y)
   len(nil) -> 0
   len(cons(x, xs)) -> s(len(xs))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))
   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   take(0, cons(y, ys)) -> y
   take(s(x), cons(y, ys)) -> take(x, ys)
   addList(x, y) -> if(le(0, min(len(x), len(y))), 0, x, y, nil)
   if(false, c, x, y, z) -> z
   if(true, c, xs, ys, z) -> if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_len(x_1)) -> len(encArg(x_1))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_len(x_1) -> len(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_addList(x_1, x_2) -> addList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))


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

(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:

   min(0, y) -> 0
   min(s(x), 0) -> 0
   min(s(x), s(y)) -> min(x, y)
   len(nil) -> 0
   len(cons(x, xs)) -> s(len(xs))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))
   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   take(0, cons(y, ys)) -> y
   take(s(x), cons(y, ys)) -> take(x, ys)
   addList(x, y) -> if(le(0, min(len(x), len(y))), 0, x, y, nil)
   if(false, c, x, y, z) -> z
   if(true, c, xs, ys, z) -> if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_len(x_1)) -> len(encArg(x_1))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_len(x_1) -> len(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_addList(x_1, x_2) -> addList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

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:

   min(0, y) -> 0
   min(s(x), 0) -> 0
   min(s(x), s(y)) -> min(x, y)
   len(nil) -> 0
   len(cons(x, xs)) -> s(len(xs))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))
   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   take(0, cons(y, ys)) -> y
   take(s(x), cons(y, ys)) -> take(x, ys)
   addList(x, y) -> if(le(0, min(len(x), len(y))), 0, x, y, nil)
   if(false, c, x, y, z) -> z
   if(true, c, xs, ys, z) -> if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_len(x_1)) -> len(encArg(x_1))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_len(x_1) -> len(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_addList(x_1, x_2) -> addList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

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:

   min(0, y) -> 0
   min(s(x), 0) -> 0
   min(s(x), s(y)) -> min(x, y)
   len(nil) -> 0
   len(cons(x, xs)) -> s(len(xs))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))
   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   take(0, cons(y, ys)) -> y
   take(s(x), cons(y, ys)) -> take(x, ys)
   addList(x, y) -> if(le(0, min(len(x), len(y))), 0, x, y, nil)
   if(false, c, x, y, z) -> z
   if(true, c, xs, ys, z) -> if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_len(x_1)) -> len(encArg(x_1))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_len(x_1) -> len(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_addList(x_1, x_2) -> addList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

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

min(s(x), s(y)) ->^+ min(x, y)

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

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

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:

   min(0, y) -> 0
   min(s(x), 0) -> 0
   min(s(x), s(y)) -> min(x, y)
   len(nil) -> 0
   len(cons(x, xs)) -> s(len(xs))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))
   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   take(0, cons(y, ys)) -> y
   take(s(x), cons(y, ys)) -> take(x, ys)
   addList(x, y) -> if(le(0, min(len(x), len(y))), 0, x, y, nil)
   if(false, c, x, y, z) -> z
   if(true, c, xs, ys, z) -> if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_len(x_1)) -> len(encArg(x_1))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_len(x_1) -> len(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_addList(x_1, x_2) -> addList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

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:

   min(0, y) -> 0
   min(s(x), 0) -> 0
   min(s(x), s(y)) -> min(x, y)
   len(nil) -> 0
   len(cons(x, xs)) -> s(len(xs))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))
   le(0, x) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   take(0, cons(y, ys)) -> y
   take(s(x), cons(y, ys)) -> take(x, ys)
   addList(x, y) -> if(le(0, min(len(x), len(y))), 0, x, y, nil)
   if(false, c, x, y, z) -> z
   if(true, c, xs, ys, z) -> if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_len(x_1)) -> len(encArg(x_1))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_len(x_1) -> len(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_addList(x_1, x_2) -> addList(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

Rewrite Strategy: INNERMOST