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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 9624 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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   MINUS(0, z0) -> c5
   MINUS(z0, 0) -> c6
   MINUS(s(z0), s(z1)) -> c7(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c8
   QUOT(s(z0), s(z1)) -> c9(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LOG(z0) -> c10(LOG2(z0, 0))
   LOG2(z0, z1) -> c11(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, 0))
   LOG2(z0, z1) -> c12(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c13(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(IF2(z0, z1, z2))
   IF2(true, z0, s(z1)) -> c16
   IF2(false, z0, z1) -> c17(LOG2(quot(z0, s(s(0))), z1), QUOT(z0, s(s(0))))

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   log(z0) -> log2(z0, 0)
   log2(z0, z1) -> if(le(z0, 0), le(z0, s(0)), z0, inc(z1))
   if(true, z0, z1, z2) -> log_undefined
   if(false, z0, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, s(z1)) -> z1
   if2(false, z0, z1) -> log2(quot(z0, s(s(0))), z1)

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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   MINUS(0, z0) -> c5
   MINUS(z0, 0) -> c6
   MINUS(s(z0), s(z1)) -> c7(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c8
   QUOT(s(z0), s(z1)) -> c9(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LOG(z0) -> c10(LOG2(z0, 0))
   LOG2(z0, z1) -> c11(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, 0))
   LOG2(z0, z1) -> c12(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c13(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(IF2(z0, z1, z2))
   IF2(true, z0, s(z1)) -> c16
   IF2(false, z0, z1) -> c17(LOG2(quot(z0, s(s(0))), z1), QUOT(z0, s(s(0))))

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   log(z0) -> log2(z0, 0)
   log2(z0, z1) -> if(le(z0, 0), le(z0, s(0)), z0, inc(z1))
   if(true, z0, z1, z2) -> log_undefined
   if(false, z0, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, s(z1)) -> z1
   if2(false, z0, z1) -> log2(quot(z0, s(s(0))), z1)

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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   MINUS(0, z0) -> c5
   MINUS(z0, 0) -> c6
   MINUS(s(z0), s(z1)) -> c7(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c8
   QUOT(s(z0), s(z1)) -> c9(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LOG(z0) -> c10(LOG2(z0, 0))
   LOG2(z0, z1) -> c11(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, 0))
   LOG2(z0, z1) -> c12(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c13(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(IF2(z0, z1, z2))
   IF2(true, z0, s(z1)) -> c16
   IF2(false, z0, z1) -> c17(LOG2(quot(z0, s(s(0))), z1), QUOT(z0, s(s(0))))

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   log(z0) -> log2(z0, 0)
   log2(z0, z1) -> if(le(z0, 0), le(z0, s(0)), z0, inc(z1))
   if(true, z0, z1, z2) -> log_undefined
   if(false, z0, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, s(z1)) -> z1
   if2(false, z0, z1) -> log2(quot(z0, s(s(0))), z1)

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

INC(s(z0)) ->^+ c4(INC(z0))

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

The pumping substitution is [z0 / 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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   MINUS(0, z0) -> c5
   MINUS(z0, 0) -> c6
   MINUS(s(z0), s(z1)) -> c7(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c8
   QUOT(s(z0), s(z1)) -> c9(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LOG(z0) -> c10(LOG2(z0, 0))
   LOG2(z0, z1) -> c11(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, 0))
   LOG2(z0, z1) -> c12(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c13(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(IF2(z0, z1, z2))
   IF2(true, z0, s(z1)) -> c16
   IF2(false, z0, z1) -> c17(LOG2(quot(z0, s(s(0))), z1), QUOT(z0, s(s(0))))

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   log(z0) -> log2(z0, 0)
   log2(z0, z1) -> if(le(z0, 0), le(z0, s(0)), z0, inc(z1))
   if(true, z0, z1, z2) -> log_undefined
   if(false, z0, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, s(z1)) -> z1
   if2(false, z0, z1) -> log2(quot(z0, s(s(0))), z1)

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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   MINUS(0, z0) -> c5
   MINUS(z0, 0) -> c6
   MINUS(s(z0), s(z1)) -> c7(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c8
   QUOT(s(z0), s(z1)) -> c9(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LOG(z0) -> c10(LOG2(z0, 0))
   LOG2(z0, z1) -> c11(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, 0))
   LOG2(z0, z1) -> c12(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c13(IF(le(z0, 0), le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(IF2(z0, z1, z2))
   IF2(true, z0, s(z1)) -> c16
   IF2(false, z0, z1) -> c17(LOG2(quot(z0, s(s(0))), z1), QUOT(z0, s(s(0))))

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   log(z0) -> log2(z0, 0)
   log2(z0, z1) -> if(le(z0, 0), le(z0, s(0)), z0, inc(z1))
   if(true, z0, z1, z2) -> log_undefined
   if(false, z0, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, s(z1)) -> z1
   if2(false, z0, z1) -> log2(quot(z0, s(s(0))), z1)

Rewrite Strategy: INNERMOST