/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 326 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:

   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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(nil) -> nil
   encArg(0) -> 0
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encode_last(x_1) -> last(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_reverse(x_1) -> reverse(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:

   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encode_last(x_1) -> last(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_reverse(x_1) -> reverse(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:

   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encode_last(x_1) -> last(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_reverse(x_1) -> reverse(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:

   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encode_last(x_1) -> last(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_reverse(x_1) -> reverse(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

last(cons(x, cons(y, xs))) ->^+ last(cons(y, xs))

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

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

The result substitution is [x / y].




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

(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:

   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encode_last(x_1) -> last(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_reverse(x_1) -> reverse(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:

   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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

   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encode_last(x_1) -> last(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_reverse(x_1) -> reverse(encArg(x_1))

Rewrite Strategy: INNERMOST