WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1093 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


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(0)
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) -> c
   CHECK(s(0)) -> c1
   CHECK(s(s(0))) -> c2
   CHECK(s(s(s(z0)))) -> c3(CHECK(s(z0)))
   HALF(0) -> c4
   HALF(s(0)) -> c5
   HALF(s(s(z0))) -> c6(HALF(z0))
   PLUS(0, z0) -> c7
   PLUS(s(z0), z1) -> c8(PLUS(z0, z1))
   TIMES(z0, z1) -> c9(TIMESITER(z0, z1, 0))
   TIMESITER(z0, z1, z2) -> c10(IF(check(z0), z0, z1, z2, plus(z2, z1)), CHECK(z0))
   TIMESITER(z0, z1, z2) -> c11(IF(check(z0), z0, z1, z2, plus(z2, z1)), PLUS(z2, z1))
   P(s(z0)) -> c12
   P(0) -> c13
   IF(zero, z0, z1, z2, z3) -> c14
   IF(odd, z0, z1, z2, z3) -> c15(TIMESITER(p(z0), z1, z3), P(z0))
   IF(even, z0, z1, z2, z3) -> c16(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c17(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z2))
   IF(even, z0, z1, z2, z3) -> c18(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c19(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(s(z2)))

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

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(z0)))) -> check(s(z0))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0)
   timesIter(z0, z1, z2) -> if(check(z0), z0, z1, z2, plus(z2, z1))
   p(s(z0)) -> z0
   p(0) -> 0
   if(zero, z0, z1, z2, z3) -> z2
   if(odd, z0, z1, z2, z3) -> timesIter(p(z0), z1, z3)
   if(even, z0, z1, z2, z3) -> plus(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2))))

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

(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) -> c
   CHECK(s(0)) -> c1
   CHECK(s(s(0))) -> c2
   CHECK(s(s(s(z0)))) -> c3(CHECK(s(z0)))
   HALF(0) -> c4
   HALF(s(0)) -> c5
   HALF(s(s(z0))) -> c6(HALF(z0))
   PLUS(0, z0) -> c7
   PLUS(s(z0), z1) -> c8(PLUS(z0, z1))
   TIMES(z0, z1) -> c9(TIMESITER(z0, z1, 0))
   TIMESITER(z0, z1, z2) -> c10(IF(check(z0), z0, z1, z2, plus(z2, z1)), CHECK(z0))
   TIMESITER(z0, z1, z2) -> c11(IF(check(z0), z0, z1, z2, plus(z2, z1)), PLUS(z2, z1))
   P(s(z0)) -> c12
   P(0) -> c13
   IF(zero, z0, z1, z2, z3) -> c14
   IF(odd, z0, z1, z2, z3) -> c15(TIMESITER(p(z0), z1, z3), P(z0))
   IF(even, z0, z1, z2, z3) -> c16(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c17(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z2))
   IF(even, z0, z1, z2, z3) -> c18(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c19(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(s(z2)))

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

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(z0)))) -> check(s(z0))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0)
   timesIter(z0, z1, z2) -> if(check(z0), z0, z1, z2, plus(z2, z1))
   p(s(z0)) -> z0
   p(0) -> 0
   if(zero, z0, z1, z2, z3) -> z2
   if(odd, z0, z1, z2, z3) -> timesIter(p(z0), z1, z3)
   if(even, z0, z1, z2, z3) -> plus(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2))))

Rewrite Strategy: INNERMOST
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(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(4)
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) -> c
   CHECK(s(0)) -> c1
   CHECK(s(s(0))) -> c2
   CHECK(s(s(s(z0)))) -> c3(CHECK(s(z0)))
   HALF(0) -> c4
   HALF(s(0)) -> c5
   HALF(s(s(z0))) -> c6(HALF(z0))
   PLUS(0, z0) -> c7
   PLUS(s(z0), z1) -> c8(PLUS(z0, z1))
   TIMES(z0, z1) -> c9(TIMESITER(z0, z1, 0))
   TIMESITER(z0, z1, z2) -> c10(IF(check(z0), z0, z1, z2, plus(z2, z1)), CHECK(z0))
   TIMESITER(z0, z1, z2) -> c11(IF(check(z0), z0, z1, z2, plus(z2, z1)), PLUS(z2, z1))
   P(s(z0)) -> c12
   P(0) -> c13
   IF(zero, z0, z1, z2, z3) -> c14
   IF(odd, z0, z1, z2, z3) -> c15(TIMESITER(p(z0), z1, z3), P(z0))
   IF(even, z0, z1, z2, z3) -> c16(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c17(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z2))
   IF(even, z0, z1, z2, z3) -> c18(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c19(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(s(z2)))

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

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(z0)))) -> check(s(z0))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0)
   timesIter(z0, z1, z2) -> if(check(z0), z0, z1, z2, plus(z2, z1))
   p(s(z0)) -> z0
   p(0) -> 0
   if(zero, z0, z1, z2, z3) -> z2
   if(odd, z0, z1, z2, z3) -> timesIter(p(z0), z1, z3)
   if(even, z0, z1, z2, z3) -> plus(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2))))

Rewrite Strategy: INNERMOST
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(5) 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(z0)))) ->^+ c3(CHECK(s(z0)))

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

The pumping substitution is [z0 / s(s(z0))].

The result substitution is [ ].




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(6)
Complex Obligation (BEST)

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(7)
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) -> c
   CHECK(s(0)) -> c1
   CHECK(s(s(0))) -> c2
   CHECK(s(s(s(z0)))) -> c3(CHECK(s(z0)))
   HALF(0) -> c4
   HALF(s(0)) -> c5
   HALF(s(s(z0))) -> c6(HALF(z0))
   PLUS(0, z0) -> c7
   PLUS(s(z0), z1) -> c8(PLUS(z0, z1))
   TIMES(z0, z1) -> c9(TIMESITER(z0, z1, 0))
   TIMESITER(z0, z1, z2) -> c10(IF(check(z0), z0, z1, z2, plus(z2, z1)), CHECK(z0))
   TIMESITER(z0, z1, z2) -> c11(IF(check(z0), z0, z1, z2, plus(z2, z1)), PLUS(z2, z1))
   P(s(z0)) -> c12
   P(0) -> c13
   IF(zero, z0, z1, z2, z3) -> c14
   IF(odd, z0, z1, z2, z3) -> c15(TIMESITER(p(z0), z1, z3), P(z0))
   IF(even, z0, z1, z2, z3) -> c16(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c17(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z2))
   IF(even, z0, z1, z2, z3) -> c18(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c19(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(s(z2)))

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

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(z0)))) -> check(s(z0))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0)
   timesIter(z0, z1, z2) -> if(check(z0), z0, z1, z2, plus(z2, z1))
   p(s(z0)) -> z0
   p(0) -> 0
   if(zero, z0, z1, z2, z3) -> z2
   if(odd, z0, z1, z2, z3) -> timesIter(p(z0), z1, z3)
   if(even, z0, z1, z2, z3) -> plus(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2))))

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(10)
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) -> c
   CHECK(s(0)) -> c1
   CHECK(s(s(0))) -> c2
   CHECK(s(s(s(z0)))) -> c3(CHECK(s(z0)))
   HALF(0) -> c4
   HALF(s(0)) -> c5
   HALF(s(s(z0))) -> c6(HALF(z0))
   PLUS(0, z0) -> c7
   PLUS(s(z0), z1) -> c8(PLUS(z0, z1))
   TIMES(z0, z1) -> c9(TIMESITER(z0, z1, 0))
   TIMESITER(z0, z1, z2) -> c10(IF(check(z0), z0, z1, z2, plus(z2, z1)), CHECK(z0))
   TIMESITER(z0, z1, z2) -> c11(IF(check(z0), z0, z1, z2, plus(z2, z1)), PLUS(z2, z1))
   P(s(z0)) -> c12
   P(0) -> c13
   IF(zero, z0, z1, z2, z3) -> c14
   IF(odd, z0, z1, z2, z3) -> c15(TIMESITER(p(z0), z1, z3), P(z0))
   IF(even, z0, z1, z2, z3) -> c16(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c17(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(z2)), HALF(z2))
   IF(even, z0, z1, z2, z3) -> c18(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(z0))
   IF(even, z0, z1, z2, z3) -> c19(PLUS(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2)))), TIMESITER(half(z0), z1, half(s(z2))), HALF(s(z2)))

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

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(z0)))) -> check(s(z0))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0)
   timesIter(z0, z1, z2) -> if(check(z0), z0, z1, z2, plus(z2, z1))
   p(s(z0)) -> z0
   p(0) -> 0
   if(zero, z0, z1, z2, z3) -> z2
   if(odd, z0, z1, z2, z3) -> timesIter(p(z0), z1, z3)
   if(even, z0, z1, z2, z3) -> plus(timesIter(half(z0), z1, half(z2)), timesIter(half(z0), z1, half(s(z2))))

Rewrite Strategy: INNERMOST