/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 268 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 4 ms]
(8) BOUNDS(INF, INF)


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(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)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

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(nil) -> nil
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2))
   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_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2))
   encode_nil -> nil


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(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)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

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(nil) -> nil
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2))
   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_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2))
   encode_nil -> nil

Rewrite Strategy: INNERMOST
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(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)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

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(nil) -> nil
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2))
   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_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2))
   encode_nil -> nil

Rewrite Strategy: INNERMOST
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(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)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

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(nil) -> nil
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_zWquot(x_1, x_2)) -> zWquot(encArg(x_1), encArg(x_2))
   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_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_zWquot(x_1, x_2) -> zWquot(encArg(x_1), encArg(x_2))
   encode_nil -> nil

Rewrite Strategy: INNERMOST
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(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)