/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, 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) InfiniteLowerBoundProof [FINISHED, 0 ms]
        (10) BOUNDS(INF, INF)


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


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

S is empty.
Rewrite Strategy: INNERMOST
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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

add(s(X), Y) ->^+ s(add(X, Y))

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

The pumping substitution is [X / s(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 (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

S is empty.
Rewrite Strategy: INNERMOST
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(9) InfiniteLowerBoundProof (FINISHED)
The following loop proves infinite runtime complexity:

The rewrite sequence

fact(X) ->^+ if(zero(X), s(0), prod(X, fact(p(X))))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].

The pumping substitution is [ ].

The result substitution is [X / p(X)].




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(10)
BOUNDS(INF, INF)