/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 302 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
        (13) DecreasingLoopProof [FINISHED, 515 ms]
        (14) 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:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_perfectp(x_1)) -> perfectp(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_perfectp(x_1) -> perfectp(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))


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

(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:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_perfectp(x_1)) -> perfectp(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_perfectp(x_1) -> perfectp(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))

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:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_perfectp(x_1)) -> perfectp(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_perfectp(x_1) -> perfectp(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))

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:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_perfectp(x_1)) -> perfectp(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_perfectp(x_1) -> perfectp(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))

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

le(s(x), s(y)) ->^+ le(x, y)

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

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

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(EXP, INF).


The TRS R consists of the following rules:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_perfectp(x_1)) -> perfectp(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_perfectp(x_1) -> perfectp(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))

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(EXP, INF).


The TRS R consists of the following rules:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_perfectp(x_1)) -> perfectp(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_perfectp(x_1) -> perfectp(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))

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

(13) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

f(s(s(x1_0)), s(y), z, s(0)) ->^+ if(le(s(x1_0), y), f(s(s(x1_0)), minus(y, s(x1_0)), z, s(0)), if(le(x1_0, 0), f(x1_0, s(0), minus(z, s(x1_0)), s(0)), f(x1_0, s(0), z, s(0))))

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

The pumping substitution is [x1_0 / s(s(x1_0))].

The result substitution is [y / 0, z / minus(z, s(x1_0))].



The rewrite sequence

f(s(s(x1_0)), s(y), z, s(0)) ->^+ if(le(s(x1_0), y), f(s(s(x1_0)), minus(y, s(x1_0)), z, s(0)), if(le(x1_0, 0), f(x1_0, s(0), minus(z, s(x1_0)), s(0)), f(x1_0, s(0), z, s(0))))

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

The pumping substitution is [x1_0 / s(s(x1_0))].

The result substitution is [y / 0].




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(14)
BOUNDS(EXP, INF)