/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection
        (9) DecreasingLoopProof [FINISHED, 23 ms]
        (10) BOUNDS(EXP, INF)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
   0 -> n__0
   s(X) -> n__s(X)
   diff(X1, X2) -> n__diff(X1, X2)
   p(X) -> n__p(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2))
   activate(n__p(X)) -> p(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(2)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
   0 -> n__0
   s(X) -> n__s(X)
   diff(X1, X2) -> n__diff(X1, X2)
   p(X) -> n__p(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2))
   activate(n__p(X)) -> p(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(3) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

activate(n__p(X)) ->^+ p(activate(X))

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

The pumping substitution is [X / n__p(X)].

The result substitution is [ ].




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(4)
Complex Obligation (BEST)

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(5)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
   0 -> n__0
   s(X) -> n__s(X)
   diff(X1, X2) -> n__diff(X1, X2)
   p(X) -> n__p(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2))
   activate(n__p(X)) -> p(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y)))
   0 -> n__0
   s(X) -> n__s(X)
   diff(X1, X2) -> n__diff(X1, X2)
   p(X) -> n__p(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2))
   activate(n__p(X)) -> p(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(9) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

activate(n__diff(X1, X2)) ->^+ if(leq(activate(X1), activate(X2)), n__0, n__s(n__diff(n__p(activate(X1)), activate(X2))))

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

The pumping substitution is [X1 / n__diff(X1, X2)].

The result substitution is [ ].



The rewrite sequence

activate(n__diff(X1, X2)) ->^+ if(leq(activate(X1), activate(X2)), n__0, n__s(n__diff(n__p(activate(X1)), activate(X2))))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2,0,0,0].

The pumping substitution is [X1 / n__diff(X1, X2)].

The result substitution is [ ].




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(10)
BOUNDS(EXP, INF)