/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(EXP, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 217 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [FINISHED, 0 ms]
(8) BOUNDS(EXP, INF)


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

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


The TRS R consists of the following rules:

   bsort(nil) -> nil
   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
   bubble(nil) -> nil
   bubble(.(x, nil)) -> .(x, nil)
   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
   last(nil) -> 0
   last(.(x, nil)) -> x
   last(.(x, .(y, z))) -> last(.(y, z))
   butlast(nil) -> nil
   butlast(.(x, nil)) -> nil
   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))

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(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_bsort(x_1)) -> bsort(encArg(x_1))
   encArg(cons_bubble(x_1)) -> bubble(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_butlast(x_1)) -> butlast(encArg(x_1))
   encode_bsort(x_1) -> bsort(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_last(x_1) -> last(encArg(x_1))
   encode_bubble(x_1) -> bubble(encArg(x_1))
   encode_butlast(x_1) -> butlast(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2))
   encode_0 -> 0


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


The TRS R consists of the following rules:

   bsort(nil) -> nil
   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
   bubble(nil) -> nil
   bubble(.(x, nil)) -> .(x, nil)
   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
   last(nil) -> 0
   last(.(x, nil)) -> x
   last(.(x, .(y, z))) -> last(.(y, z))
   butlast(nil) -> nil
   butlast(.(x, nil)) -> nil
   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_bsort(x_1)) -> bsort(encArg(x_1))
   encArg(cons_bubble(x_1)) -> bubble(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_butlast(x_1)) -> butlast(encArg(x_1))
   encode_bsort(x_1) -> bsort(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_last(x_1) -> last(encArg(x_1))
   encode_bubble(x_1) -> bubble(encArg(x_1))
   encode_butlast(x_1) -> butlast(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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(EXP, INF).


The TRS R consists of the following rules:

   bsort(nil) -> nil
   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
   bubble(nil) -> nil
   bubble(.(x, nil)) -> .(x, nil)
   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
   last(nil) -> 0
   last(.(x, nil)) -> x
   last(.(x, .(y, z))) -> last(.(y, z))
   butlast(nil) -> nil
   butlast(.(x, nil)) -> nil
   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_bsort(x_1)) -> bsort(encArg(x_1))
   encArg(cons_bubble(x_1)) -> bubble(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_butlast(x_1)) -> butlast(encArg(x_1))
   encode_bsort(x_1) -> bsort(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_last(x_1) -> last(encArg(x_1))
   encode_bubble(x_1) -> bubble(encArg(x_1))
   encode_butlast(x_1) -> butlast(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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(EXP, INF).


The TRS R consists of the following rules:

   bsort(nil) -> nil
   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
   bubble(nil) -> nil
   bubble(.(x, nil)) -> .(x, nil)
   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
   last(nil) -> 0
   last(.(x, nil)) -> x
   last(.(x, .(y, z))) -> last(.(y, z))
   butlast(nil) -> nil
   butlast(.(x, nil)) -> nil
   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_bsort(x_1)) -> bsort(encArg(x_1))
   encArg(cons_bubble(x_1)) -> bubble(encArg(x_1))
   encArg(cons_last(x_1)) -> last(encArg(x_1))
   encArg(cons_butlast(x_1)) -> butlast(encArg(x_1))
   encode_bsort(x_1) -> bsort(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_last(x_1) -> last(encArg(x_1))
   encode_bubble(x_1) -> bubble(encArg(x_1))
   encode_butlast(x_1) -> butlast(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST
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(7) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

bubble(.(x, .(y, z))) ->^+ if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))

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

The pumping substitution is [z / .(y, z)].

The result substitution is [ ].



The rewrite sequence

bubble(.(x, .(y, z))) ->^+ if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].

The pumping substitution is [z / .(y, z)].

The result substitution is [x / y].




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(8)
BOUNDS(EXP, INF)