WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 1289 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 8 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:

   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c3
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c6
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUM(z0) -> c8(SUMITER(z0, 0))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c11
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
   ISEMPTY(nil) -> c13
   ISEMPTY(cons(z0, z1)) -> c14
   HEAD(nil) -> c15
   HEAD(cons(z0, z1)) -> c16
   TAIL(nil) -> c17
   TAIL(cons(z0, z1)) -> c18
   A -> c19
   A -> c20

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

   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c

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

   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c3
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c6
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUM(z0) -> c8(SUMITER(z0, 0))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c11
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
   ISEMPTY(nil) -> c13
   ISEMPTY(cons(z0, z1)) -> c14
   HEAD(nil) -> c15
   HEAD(cons(z0, z1)) -> c16
   TAIL(nil) -> c17
   TAIL(cons(z0, z1)) -> c18
   A -> c19
   A -> c20

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

   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c

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

   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c3
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c6
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUM(z0) -> c8(SUMITER(z0, 0))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c11
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
   ISEMPTY(nil) -> c13
   ISEMPTY(cons(z0, z1)) -> c14
   HEAD(nil) -> c15
   HEAD(cons(z0, z1)) -> c16
   TAIL(nil) -> c17
   TAIL(cons(z0, z1)) -> c18
   A -> c19
   A -> c20

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

   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c

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

LE(s(z0), s(z1)) ->^+ c7(LE(z0, z1))

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

The pumping substitution is [z0 / s(z0), z1 / s(z1)].

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:

   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c3
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c6
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUM(z0) -> c8(SUMITER(z0, 0))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c11
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
   ISEMPTY(nil) -> c13
   ISEMPTY(cons(z0, z1)) -> c14
   HEAD(nil) -> c15
   HEAD(cons(z0, z1)) -> c16
   TAIL(nil) -> c17
   TAIL(cons(z0, z1)) -> c18
   A -> c19
   A -> c20

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

   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c

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:

   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c3
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c6
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUM(z0) -> c8(SUMITER(z0, 0))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c11
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
   ISEMPTY(nil) -> c13
   ISEMPTY(cons(z0, z1)) -> c14
   HEAD(nil) -> c15
   HEAD(cons(z0, z1)) -> c16
   TAIL(nil) -> c17
   TAIL(cons(z0, z1)) -> c18
   A -> c19
   A -> c20

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

   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c

Rewrite Strategy: INNERMOST