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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 1065 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:

   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(0) -> c3(FIB(0))
   FIBO(s(0)) -> c4(FIB(s(0)))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   FIB(0) -> c7
   FIB(s(0)) -> c8
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, 0) -> c13
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))

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

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, 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:

   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(0) -> c3(FIB(0))
   FIBO(s(0)) -> c4(FIB(s(0)))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   FIB(0) -> c7
   FIB(s(0)) -> c8
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, 0) -> c13
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))

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

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, 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:

   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(0) -> c3(FIB(0))
   FIBO(s(0)) -> c4(FIB(s(0)))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   FIB(0) -> c7
   FIB(s(0)) -> c8
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, 0) -> c13
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))

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

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, 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

SUM(z0, s(z1)) ->^+ c14(SUM(z0, z1))

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

The pumping substitution is [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:

   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(0) -> c3(FIB(0))
   FIBO(s(0)) -> c4(FIB(s(0)))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   FIB(0) -> c7
   FIB(s(0)) -> c8
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, 0) -> c13
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))

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

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, 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:

   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(0) -> c3(FIB(0))
   FIBO(s(0)) -> c4(FIB(s(0)))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   FIB(0) -> c7
   FIB(s(0)) -> c8
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, 0) -> c13
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))

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

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))

Rewrite Strategy: INNERMOST