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


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WORST_CASE(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(INF, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 492 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 5 ms]
(8) BOUNDS(INF, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(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(INF, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(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(INF, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) InfiniteLowerBoundProof (FINISHED)
The following loop proves infinite runtime complexity:

The rewrite sequence

from(X) ->^+ cons(X, from(s(X)))

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

The pumping substitution is [ ].

The result substitution is [X / s(X)].




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(8)
BOUNDS(INF, INF)