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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 297 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:

   isList(nil) -> tt
   isList(Cons(x, xs)) -> isList(xs)
   downfrom(0) -> nil
   downfrom(s(x)) -> Cons(s(x), downfrom(x))
   f(x) -> cond(isList(downfrom(x)), s(x))
   cond(tt, x) -> f(x)

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(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encArg(cons_downfrom(x_1)) -> downfrom(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_nil -> nil
   encode_tt -> tt
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_downfrom(x_1) -> downfrom(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))


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

(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:

   isList(nil) -> tt
   isList(Cons(x, xs)) -> isList(xs)
   downfrom(0) -> nil
   downfrom(s(x)) -> Cons(s(x), downfrom(x))
   f(x) -> cond(isList(downfrom(x)), s(x))
   cond(tt, x) -> f(x)

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encArg(cons_downfrom(x_1)) -> downfrom(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_nil -> nil
   encode_tt -> tt
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_downfrom(x_1) -> downfrom(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))

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:

   isList(nil) -> tt
   isList(Cons(x, xs)) -> isList(xs)
   downfrom(0) -> nil
   downfrom(s(x)) -> Cons(s(x), downfrom(x))
   f(x) -> cond(isList(downfrom(x)), s(x))
   cond(tt, x) -> f(x)

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encArg(cons_downfrom(x_1)) -> downfrom(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_nil -> nil
   encode_tt -> tt
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_downfrom(x_1) -> downfrom(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))

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:

   isList(nil) -> tt
   isList(Cons(x, xs)) -> isList(xs)
   downfrom(0) -> nil
   downfrom(s(x)) -> Cons(s(x), downfrom(x))
   f(x) -> cond(isList(downfrom(x)), s(x))
   cond(tt, x) -> f(x)

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encArg(cons_downfrom(x_1)) -> downfrom(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_nil -> nil
   encode_tt -> tt
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_downfrom(x_1) -> downfrom(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))

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

downfrom(s(x)) ->^+ Cons(s(x), downfrom(x))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

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:

   isList(nil) -> tt
   isList(Cons(x, xs)) -> isList(xs)
   downfrom(0) -> nil
   downfrom(s(x)) -> Cons(s(x), downfrom(x))
   f(x) -> cond(isList(downfrom(x)), s(x))
   cond(tt, x) -> f(x)

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encArg(cons_downfrom(x_1)) -> downfrom(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_nil -> nil
   encode_tt -> tt
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_downfrom(x_1) -> downfrom(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))

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:

   isList(nil) -> tt
   isList(Cons(x, xs)) -> isList(xs)
   downfrom(0) -> nil
   downfrom(s(x)) -> Cons(s(x), downfrom(x))
   f(x) -> cond(isList(downfrom(x)), s(x))
   cond(tt, x) -> f(x)

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

   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encArg(cons_downfrom(x_1)) -> downfrom(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_nil -> nil
   encode_tt -> tt
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_downfrom(x_1) -> downfrom(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST