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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 3753 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 9 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:

   DIV(z0, z1) -> c1(DIV2(z0, z1, 0))
   DIV2(z0, z1, z2) -> c2(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, 0))
   DIV2(z0, z1, z2) -> c3(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, z0))
   DIV2(z0, z1, z2) -> c4(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), PLUS(z2, 0))
   DIV2(z0, z1, z2) -> c5(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), INC(z2))
   IF1(true, z0, z1, z2, z3, z4) -> c6
   IF1(false, z0, z1, z2, z3, z4) -> c7(IF2(z0, z1, z2, z3, z4))
   IF2(true, z0, z1, z2, z3) -> c8(DIV2(minus(z0, z1), z1, z3), MINUS(z0, z1))
   IF2(false, z0, z1, z2, z3) -> c9
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
   LE(s(z0), 0) -> c12
   LE(0, z0) -> c13
   LE(s(z0), s(z1)) -> c14(LE(z0, z1))
   MINUS(z0, 0) -> c15
   MINUS(0, z0) -> c16
   MINUS(s(z0), s(z1)) -> c17(MINUS(z0, z1))
   PLUS(z0, z1) -> c18(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c19(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c20
   IFPLUS(false, z0, z1, z2) -> c21(PLUSITER(z0, s(z1), s(z2)))
   A -> c22
   A -> c23

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

   div(z0, z1) -> div2(z0, z1, 0)
   div2(z0, z1, z2) -> if1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2))
   if1(true, z0, z1, z2, z3, z4) -> divZeroError
   if1(false, z0, z1, z2, z3, z4) -> if2(z0, z1, z2, z3, z4)
   if2(true, z0, z1, z2, z3) -> div2(minus(z0, z1), z1, z3)
   if2(false, z0, z1, z2, z3) -> z2
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   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))
   a -> c
   a -> d

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:

   DIV(z0, z1) -> c1(DIV2(z0, z1, 0))
   DIV2(z0, z1, z2) -> c2(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, 0))
   DIV2(z0, z1, z2) -> c3(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, z0))
   DIV2(z0, z1, z2) -> c4(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), PLUS(z2, 0))
   DIV2(z0, z1, z2) -> c5(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), INC(z2))
   IF1(true, z0, z1, z2, z3, z4) -> c6
   IF1(false, z0, z1, z2, z3, z4) -> c7(IF2(z0, z1, z2, z3, z4))
   IF2(true, z0, z1, z2, z3) -> c8(DIV2(minus(z0, z1), z1, z3), MINUS(z0, z1))
   IF2(false, z0, z1, z2, z3) -> c9
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
   LE(s(z0), 0) -> c12
   LE(0, z0) -> c13
   LE(s(z0), s(z1)) -> c14(LE(z0, z1))
   MINUS(z0, 0) -> c15
   MINUS(0, z0) -> c16
   MINUS(s(z0), s(z1)) -> c17(MINUS(z0, z1))
   PLUS(z0, z1) -> c18(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c19(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c20
   IFPLUS(false, z0, z1, z2) -> c21(PLUSITER(z0, s(z1), s(z2)))
   A -> c22
   A -> c23

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

   div(z0, z1) -> div2(z0, z1, 0)
   div2(z0, z1, z2) -> if1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2))
   if1(true, z0, z1, z2, z3, z4) -> divZeroError
   if1(false, z0, z1, z2, z3, z4) -> if2(z0, z1, z2, z3, z4)
   if2(true, z0, z1, z2, z3) -> div2(minus(z0, z1), z1, z3)
   if2(false, z0, z1, z2, z3) -> z2
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   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))
   a -> c
   a -> d

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:

   DIV(z0, z1) -> c1(DIV2(z0, z1, 0))
   DIV2(z0, z1, z2) -> c2(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, 0))
   DIV2(z0, z1, z2) -> c3(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, z0))
   DIV2(z0, z1, z2) -> c4(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), PLUS(z2, 0))
   DIV2(z0, z1, z2) -> c5(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), INC(z2))
   IF1(true, z0, z1, z2, z3, z4) -> c6
   IF1(false, z0, z1, z2, z3, z4) -> c7(IF2(z0, z1, z2, z3, z4))
   IF2(true, z0, z1, z2, z3) -> c8(DIV2(minus(z0, z1), z1, z3), MINUS(z0, z1))
   IF2(false, z0, z1, z2, z3) -> c9
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
   LE(s(z0), 0) -> c12
   LE(0, z0) -> c13
   LE(s(z0), s(z1)) -> c14(LE(z0, z1))
   MINUS(z0, 0) -> c15
   MINUS(0, z0) -> c16
   MINUS(s(z0), s(z1)) -> c17(MINUS(z0, z1))
   PLUS(z0, z1) -> c18(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c19(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c20
   IFPLUS(false, z0, z1, z2) -> c21(PLUSITER(z0, s(z1), s(z2)))
   A -> c22
   A -> c23

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

   div(z0, z1) -> div2(z0, z1, 0)
   div2(z0, z1, z2) -> if1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2))
   if1(true, z0, z1, z2, z3, z4) -> divZeroError
   if1(false, z0, z1, z2, z3, z4) -> if2(z0, z1, z2, z3, z4)
   if2(true, z0, z1, z2, z3) -> div2(minus(z0, z1), z1, z3)
   if2(false, z0, z1, z2, z3) -> z2
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   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))
   a -> c
   a -> d

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

MINUS(s(z0), s(z1)) ->^+ c17(MINUS(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:

   DIV(z0, z1) -> c1(DIV2(z0, z1, 0))
   DIV2(z0, z1, z2) -> c2(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, 0))
   DIV2(z0, z1, z2) -> c3(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, z0))
   DIV2(z0, z1, z2) -> c4(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), PLUS(z2, 0))
   DIV2(z0, z1, z2) -> c5(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), INC(z2))
   IF1(true, z0, z1, z2, z3, z4) -> c6
   IF1(false, z0, z1, z2, z3, z4) -> c7(IF2(z0, z1, z2, z3, z4))
   IF2(true, z0, z1, z2, z3) -> c8(DIV2(minus(z0, z1), z1, z3), MINUS(z0, z1))
   IF2(false, z0, z1, z2, z3) -> c9
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
   LE(s(z0), 0) -> c12
   LE(0, z0) -> c13
   LE(s(z0), s(z1)) -> c14(LE(z0, z1))
   MINUS(z0, 0) -> c15
   MINUS(0, z0) -> c16
   MINUS(s(z0), s(z1)) -> c17(MINUS(z0, z1))
   PLUS(z0, z1) -> c18(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c19(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c20
   IFPLUS(false, z0, z1, z2) -> c21(PLUSITER(z0, s(z1), s(z2)))
   A -> c22
   A -> c23

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

   div(z0, z1) -> div2(z0, z1, 0)
   div2(z0, z1, z2) -> if1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2))
   if1(true, z0, z1, z2, z3, z4) -> divZeroError
   if1(false, z0, z1, z2, z3, z4) -> if2(z0, z1, z2, z3, z4)
   if2(true, z0, z1, z2, z3) -> div2(minus(z0, z1), z1, z3)
   if2(false, z0, z1, z2, z3) -> z2
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   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))
   a -> c
   a -> d

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:

   DIV(z0, z1) -> c1(DIV2(z0, z1, 0))
   DIV2(z0, z1, z2) -> c2(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, 0))
   DIV2(z0, z1, z2) -> c3(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), LE(z1, z0))
   DIV2(z0, z1, z2) -> c4(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), PLUS(z2, 0))
   DIV2(z0, z1, z2) -> c5(IF1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2)), INC(z2))
   IF1(true, z0, z1, z2, z3, z4) -> c6
   IF1(false, z0, z1, z2, z3, z4) -> c7(IF2(z0, z1, z2, z3, z4))
   IF2(true, z0, z1, z2, z3) -> c8(DIV2(minus(z0, z1), z1, z3), MINUS(z0, z1))
   IF2(false, z0, z1, z2, z3) -> c9
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
   LE(s(z0), 0) -> c12
   LE(0, z0) -> c13
   LE(s(z0), s(z1)) -> c14(LE(z0, z1))
   MINUS(z0, 0) -> c15
   MINUS(0, z0) -> c16
   MINUS(s(z0), s(z1)) -> c17(MINUS(z0, z1))
   PLUS(z0, z1) -> c18(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c19(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c20
   IFPLUS(false, z0, z1, z2) -> c21(PLUSITER(z0, s(z1), s(z2)))
   A -> c22
   A -> c23

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

   div(z0, z1) -> div2(z0, z1, 0)
   div2(z0, z1, z2) -> if1(le(z1, 0), le(z1, z0), z0, z1, plus(z2, 0), inc(z2))
   if1(true, z0, z1, z2, z3, z4) -> divZeroError
   if1(false, z0, z1, z2, z3, z4) -> if2(z0, z1, z2, z3, z4)
   if2(true, z0, z1, z2, z3) -> div2(minus(z0, z1), z1, z3)
   if2(false, z0, z1, z2, z3) -> z2
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   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))
   a -> c
   a -> d

Rewrite Strategy: INNERMOST