/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(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 254 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


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(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:

   0(#) -> #
   +(#, x) -> x
   +(x, #) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), j(y)) -> j(+(x, y))
   +(j(x), 0(y)) -> j(+(x, y))
   +(1(x), 1(y)) -> j(+(+(x, y), 1(#)))
   +(j(x), j(y)) -> 1(+(+(x, y), j(#)))
   +(1(x), j(y)) -> 0(+(x, y))
   +(j(x), 1(y)) -> 0(+(x, y))
   +(+(x, y), z) -> +(x, +(y, z))
   opp(#) -> #
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   -(x, y) -> +(x, opp(y))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   *(j(x), y) -> -(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))

S is empty.
Rewrite Strategy: INNERMOST
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(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(#) -> #
   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(j(x_1)) -> j(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_opp(x_1)) -> opp(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_# -> #
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_opp(x_1) -> opp(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))


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(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:

   0(#) -> #
   +(#, x) -> x
   +(x, #) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), j(y)) -> j(+(x, y))
   +(j(x), 0(y)) -> j(+(x, y))
   +(1(x), 1(y)) -> j(+(+(x, y), 1(#)))
   +(j(x), j(y)) -> 1(+(+(x, y), j(#)))
   +(1(x), j(y)) -> 0(+(x, y))
   +(j(x), 1(y)) -> 0(+(x, y))
   +(+(x, y), z) -> +(x, +(y, z))
   opp(#) -> #
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   -(x, y) -> +(x, opp(y))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   *(j(x), y) -> -(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))

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

   encArg(#) -> #
   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(j(x_1)) -> j(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_opp(x_1)) -> opp(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_# -> #
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_opp(x_1) -> opp(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

   0(#) -> #
   +(#, x) -> x
   +(x, #) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), j(y)) -> j(+(x, y))
   +(j(x), 0(y)) -> j(+(x, y))
   +(1(x), 1(y)) -> j(+(+(x, y), 1(#)))
   +(j(x), j(y)) -> 1(+(+(x, y), j(#)))
   +(1(x), j(y)) -> 0(+(x, y))
   +(j(x), 1(y)) -> 0(+(x, y))
   +(+(x, y), z) -> +(x, +(y, z))
   opp(#) -> #
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   -(x, y) -> +(x, opp(y))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   *(j(x), y) -> -(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))

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

   encArg(#) -> #
   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(j(x_1)) -> j(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_opp(x_1)) -> opp(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_# -> #
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_opp(x_1) -> opp(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))

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(n^1, INF).


The TRS R consists of the following rules:

   0(#) -> #
   +(#, x) -> x
   +(x, #) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), j(y)) -> j(+(x, y))
   +(j(x), 0(y)) -> j(+(x, y))
   +(1(x), 1(y)) -> j(+(+(x, y), 1(#)))
   +(j(x), j(y)) -> 1(+(+(x, y), j(#)))
   +(1(x), j(y)) -> 0(+(x, y))
   +(j(x), 1(y)) -> 0(+(x, y))
   +(+(x, y), z) -> +(x, +(y, z))
   opp(#) -> #
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   -(x, y) -> +(x, opp(y))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   *(j(x), y) -> -(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))

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

   encArg(#) -> #
   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(j(x_1)) -> j(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_opp(x_1)) -> opp(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_# -> #
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_opp(x_1) -> opp(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

+(j(x), j(y)) ->^+ 1(+(+(x, y), j(#)))

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

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

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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(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:

   0(#) -> #
   +(#, x) -> x
   +(x, #) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), j(y)) -> j(+(x, y))
   +(j(x), 0(y)) -> j(+(x, y))
   +(1(x), 1(y)) -> j(+(+(x, y), 1(#)))
   +(j(x), j(y)) -> 1(+(+(x, y), j(#)))
   +(1(x), j(y)) -> 0(+(x, y))
   +(j(x), 1(y)) -> 0(+(x, y))
   +(+(x, y), z) -> +(x, +(y, z))
   opp(#) -> #
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   -(x, y) -> +(x, opp(y))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   *(j(x), y) -> -(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))

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

   encArg(#) -> #
   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(j(x_1)) -> j(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_opp(x_1)) -> opp(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_# -> #
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_opp(x_1) -> opp(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(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:

   0(#) -> #
   +(#, x) -> x
   +(x, #) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), j(y)) -> j(+(x, y))
   +(j(x), 0(y)) -> j(+(x, y))
   +(1(x), 1(y)) -> j(+(+(x, y), 1(#)))
   +(j(x), j(y)) -> 1(+(+(x, y), j(#)))
   +(1(x), j(y)) -> 0(+(x, y))
   +(j(x), 1(y)) -> 0(+(x, y))
   +(+(x, y), z) -> +(x, +(y, z))
   opp(#) -> #
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   -(x, y) -> +(x, opp(y))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   *(j(x), y) -> -(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))

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

   encArg(#) -> #
   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(j(x_1)) -> j(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_opp(x_1)) -> opp(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_# -> #
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_opp(x_1) -> opp(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST