/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 463 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:

   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
   terms(X) -> n__terms(X)
   s(X) -> n__s(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> 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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(recip(x_1)) -> recip(encArg(x_1))
   encArg(n__terms(x_1)) -> n__terms(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(cons_terms(x_1)) -> terms(encArg(x_1))
   encArg(cons_sqr(x_1)) -> sqr(encArg(x_1))
   encArg(cons_dbl(x_1)) -> dbl(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_terms(x_1) -> terms(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_recip(x_1) -> recip(encArg(x_1))
   encode_sqr(x_1) -> sqr(encArg(x_1))
   encode_n__terms(x_1) -> n__terms(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_dbl(x_1) -> dbl(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_half(x_1) -> half(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:

   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
   terms(X) -> n__terms(X)
   s(X) -> n__s(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(recip(x_1)) -> recip(encArg(x_1))
   encArg(n__terms(x_1)) -> n__terms(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(cons_terms(x_1)) -> terms(encArg(x_1))
   encArg(cons_sqr(x_1)) -> sqr(encArg(x_1))
   encArg(cons_dbl(x_1)) -> dbl(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_terms(x_1) -> terms(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_recip(x_1) -> recip(encArg(x_1))
   encode_sqr(x_1) -> sqr(encArg(x_1))
   encode_n__terms(x_1) -> n__terms(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_dbl(x_1) -> dbl(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_half(x_1) -> half(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:

   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
   terms(X) -> n__terms(X)
   s(X) -> n__s(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(recip(x_1)) -> recip(encArg(x_1))
   encArg(n__terms(x_1)) -> n__terms(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(cons_terms(x_1)) -> terms(encArg(x_1))
   encArg(cons_sqr(x_1)) -> sqr(encArg(x_1))
   encArg(cons_dbl(x_1)) -> dbl(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_terms(x_1) -> terms(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_recip(x_1) -> recip(encArg(x_1))
   encode_sqr(x_1) -> sqr(encArg(x_1))
   encode_n__terms(x_1) -> n__terms(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_dbl(x_1) -> dbl(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_half(x_1) -> half(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:

   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
   terms(X) -> n__terms(X)
   s(X) -> n__s(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(recip(x_1)) -> recip(encArg(x_1))
   encArg(n__terms(x_1)) -> n__terms(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(cons_terms(x_1)) -> terms(encArg(x_1))
   encArg(cons_sqr(x_1)) -> sqr(encArg(x_1))
   encArg(cons_dbl(x_1)) -> dbl(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_terms(x_1) -> terms(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_recip(x_1) -> recip(encArg(x_1))
   encode_sqr(x_1) -> sqr(encArg(x_1))
   encode_n__terms(x_1) -> n__terms(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_dbl(x_1) -> dbl(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_half(x_1) -> half(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

activate(n__terms(X)) ->^+ terms(activate(X))

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

The pumping substitution is [X / n__terms(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:

   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
   terms(X) -> n__terms(X)
   s(X) -> n__s(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(recip(x_1)) -> recip(encArg(x_1))
   encArg(n__terms(x_1)) -> n__terms(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(cons_terms(x_1)) -> terms(encArg(x_1))
   encArg(cons_sqr(x_1)) -> sqr(encArg(x_1))
   encArg(cons_dbl(x_1)) -> dbl(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_terms(x_1) -> terms(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_recip(x_1) -> recip(encArg(x_1))
   encode_sqr(x_1) -> sqr(encArg(x_1))
   encode_n__terms(x_1) -> n__terms(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_dbl(x_1) -> dbl(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_half(x_1) -> half(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:

   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
   terms(X) -> n__terms(X)
   s(X) -> n__s(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(recip(x_1)) -> recip(encArg(x_1))
   encArg(n__terms(x_1)) -> n__terms(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(cons_terms(x_1)) -> terms(encArg(x_1))
   encArg(cons_sqr(x_1)) -> sqr(encArg(x_1))
   encArg(cons_dbl(x_1)) -> dbl(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_terms(x_1) -> terms(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_recip(x_1) -> recip(encArg(x_1))
   encode_sqr(x_1) -> sqr(encArg(x_1))
   encode_n__terms(x_1) -> n__terms(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_dbl(x_1) -> dbl(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))

Rewrite Strategy: INNERMOST