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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 789 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


----------------------------------------

(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), z0, round(s(z1))), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), z0, round(s(z1))), ROUND(s(z1)))
   ROUND(0) -> c2
   ROUND(s(0)) -> c3
   ROUND(s(s(z0))) -> c4(ROUND(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))

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

   f(true, z0, z1) -> f(gt(z0, z1), z0, round(s(z1)))
   round(0) -> 0
   round(s(0)) -> s(s(0))
   round(s(s(z0))) -> s(s(round(z0)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), z0, round(s(z1))), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), z0, round(s(z1))), ROUND(s(z1)))
   ROUND(0) -> c2
   ROUND(s(0)) -> c3
   ROUND(s(s(z0))) -> c4(ROUND(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))

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

   f(true, z0, z1) -> f(gt(z0, z1), z0, round(s(z1)))
   round(0) -> 0
   round(s(0)) -> s(s(0))
   round(s(s(z0))) -> s(s(round(z0)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), z0, round(s(z1))), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), z0, round(s(z1))), ROUND(s(z1)))
   ROUND(0) -> c2
   ROUND(s(0)) -> c3
   ROUND(s(s(z0))) -> c4(ROUND(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))

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

   f(true, z0, z1) -> f(gt(z0, z1), z0, round(s(z1)))
   round(0) -> 0
   round(s(0)) -> s(s(0))
   round(s(s(z0))) -> s(s(round(z0)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(5) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

ROUND(s(s(z0))) ->^+ c4(ROUND(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 [ ].




----------------------------------------

(6)
Complex Obligation (BEST)

----------------------------------------

(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), z0, round(s(z1))), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), z0, round(s(z1))), ROUND(s(z1)))
   ROUND(0) -> c2
   ROUND(s(0)) -> c3
   ROUND(s(s(z0))) -> c4(ROUND(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))

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

   f(true, z0, z1) -> f(gt(z0, z1), z0, round(s(z1)))
   round(0) -> 0
   round(s(0)) -> s(s(0))
   round(s(s(z0))) -> s(s(round(z0)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(9)
BOUNDS(n^1, INF)

----------------------------------------

(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:

   F(true, z0, z1) -> c(F(gt(z0, z1), z0, round(s(z1))), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), z0, round(s(z1))), ROUND(s(z1)))
   ROUND(0) -> c2
   ROUND(s(0)) -> c3
   ROUND(s(s(z0))) -> c4(ROUND(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))

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

   f(true, z0, z1) -> f(gt(z0, z1), z0, round(s(z1)))
   round(0) -> 0
   round(s(0)) -> s(s(0))
   round(s(s(z0))) -> s(s(round(z0)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST