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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 624 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:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(true) -> true
   encArg(undefined) -> undefined
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encArg(cons_rev(x_1, x_2, x_3, x_4)) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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_length(x_1) -> length(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_head(x_1) -> head(encArg(x_1))
   encode_undefined -> undefined
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_reverse(x_1) -> reverse(encArg(x_1))
   encode_rev(x_1, x_2, x_3, x_4) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(true) -> true
   encArg(undefined) -> undefined
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encArg(cons_rev(x_1, x_2, x_3, x_4)) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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_length(x_1) -> length(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_head(x_1) -> head(encArg(x_1))
   encode_undefined -> undefined
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_reverse(x_1) -> reverse(encArg(x_1))
   encode_rev(x_1, x_2, x_3, x_4) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(true) -> true
   encArg(undefined) -> undefined
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encArg(cons_rev(x_1, x_2, x_3, x_4)) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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_length(x_1) -> length(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_head(x_1) -> head(encArg(x_1))
   encode_undefined -> undefined
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_reverse(x_1) -> reverse(encArg(x_1))
   encode_rev(x_1, x_2, x_3, x_4) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(true) -> true
   encArg(undefined) -> undefined
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encArg(cons_rev(x_1, x_2, x_3, x_4)) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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_length(x_1) -> length(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_head(x_1) -> head(encArg(x_1))
   encode_undefined -> undefined
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_reverse(x_1) -> reverse(encArg(x_1))
   encode_rev(x_1, x_2, x_3, x_4) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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

lt(s(x), s(y)) ->^+ lt(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:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(true) -> true
   encArg(undefined) -> undefined
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encArg(cons_rev(x_1, x_2, x_3, x_4)) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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_length(x_1) -> length(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_head(x_1) -> head(encArg(x_1))
   encode_undefined -> undefined
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_reverse(x_1) -> reverse(encArg(x_1))
   encode_rev(x_1, x_2, x_3, x_4) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(true) -> true
   encArg(undefined) -> undefined
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_reverse(x_1)) -> reverse(encArg(x_1))
   encArg(cons_rev(x_1, x_2, x_3, x_4)) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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_length(x_1) -> length(encArg(x_1))
   encode_nil -> nil
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_head(x_1) -> head(encArg(x_1))
   encode_undefined -> undefined
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_reverse(x_1) -> reverse(encArg(x_1))
   encode_rev(x_1, x_2, x_3, x_4) -> rev(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   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