/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(?, O(1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, 1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
(4) BOUNDS(1, 1)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

   f(s(x)) -> f(g(x, x))
   g(0, 1) -> s(0)
   0 -> 1

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of f: g
The following defined symbols can occur below the 0th argument of g: g
The following defined symbols can occur below the 1th argument of g: g

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
g(0, 1) -> s(0)

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

   f(s(x)) -> f(g(x, x))
   0 -> 1

S is empty.
Rewrite Strategy: FULL
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(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
Constant runtime complexity proven by termination of constructor-based narrowing.

The maximal most general narrowing sequences give rise to the following rewrite sequences:

0 ->^* 1

f(s(x0)) ->^* f(g(x0, x0))




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(4)
BOUNDS(1, 1)