/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 290 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 1 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


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

(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:

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0)
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0) -> 0
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(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(0) -> 0
   encArg(zero) -> zero
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(odd) -> odd
   encArg(even) -> even
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_half(x_1)) -> half(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_timesIter(x_1, x_2, x_3)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_check(x_1) -> check(encArg(x_1))
   encode_0 -> 0
   encode_zero -> zero
   encode_s(x_1) -> s(encArg(x_1))
   encode_odd -> odd
   encode_even -> even
   encode_half(x_1) -> half(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_timesIter(x_1, x_2, x_3) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_p(x_1) -> p(encArg(x_1))


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

(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:

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0)
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0) -> 0
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

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

   encArg(0) -> 0
   encArg(zero) -> zero
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(odd) -> odd
   encArg(even) -> even
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_half(x_1)) -> half(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_timesIter(x_1, x_2, x_3)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_check(x_1) -> check(encArg(x_1))
   encode_0 -> 0
   encode_zero -> zero
   encode_s(x_1) -> s(encArg(x_1))
   encode_odd -> odd
   encode_even -> even
   encode_half(x_1) -> half(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_timesIter(x_1, x_2, x_3) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_p(x_1) -> p(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(n^1, INF).


The TRS R consists of the following rules:

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0)
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0) -> 0
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

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

   encArg(0) -> 0
   encArg(zero) -> zero
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(odd) -> odd
   encArg(even) -> even
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_half(x_1)) -> half(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_timesIter(x_1, x_2, x_3)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_check(x_1) -> check(encArg(x_1))
   encode_0 -> 0
   encode_zero -> zero
   encode_s(x_1) -> s(encArg(x_1))
   encode_odd -> odd
   encode_even -> even
   encode_half(x_1) -> half(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_timesIter(x_1, x_2, x_3) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_p(x_1) -> p(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(n^1, INF).


The TRS R consists of the following rules:

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0)
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0) -> 0
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

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

   encArg(0) -> 0
   encArg(zero) -> zero
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(odd) -> odd
   encArg(even) -> even
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_half(x_1)) -> half(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_timesIter(x_1, x_2, x_3)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_check(x_1) -> check(encArg(x_1))
   encode_0 -> 0
   encode_zero -> zero
   encode_s(x_1) -> s(encArg(x_1))
   encode_odd -> odd
   encode_even -> even
   encode_half(x_1) -> half(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_timesIter(x_1, x_2, x_3) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_p(x_1) -> p(encArg(x_1))

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

check(s(s(s(x)))) ->^+ check(s(x))

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

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

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(n^1, INF).


The TRS R consists of the following rules:

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0)
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0) -> 0
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

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

   encArg(0) -> 0
   encArg(zero) -> zero
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(odd) -> odd
   encArg(even) -> even
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_half(x_1)) -> half(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_timesIter(x_1, x_2, x_3)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_check(x_1) -> check(encArg(x_1))
   encode_0 -> 0
   encode_zero -> zero
   encode_s(x_1) -> s(encArg(x_1))
   encode_odd -> odd
   encode_even -> even
   encode_half(x_1) -> half(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_timesIter(x_1, x_2, x_3) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_p(x_1) -> p(encArg(x_1))

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

(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:

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0)
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0) -> 0
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

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

   encArg(0) -> 0
   encArg(zero) -> zero
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(odd) -> odd
   encArg(even) -> even
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_half(x_1)) -> half(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_timesIter(x_1, x_2, x_3)) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_check(x_1) -> check(encArg(x_1))
   encode_0 -> 0
   encode_zero -> zero
   encode_s(x_1) -> s(encArg(x_1))
   encode_odd -> odd
   encode_even -> even
   encode_half(x_1) -> half(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_timesIter(x_1, x_2, x_3) -> timesIter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_p(x_1) -> p(encArg(x_1))

Rewrite Strategy: INNERMOST